'«r**i*>.; '/'/'A/i'A'yy^y//i'A//^/>y/y^/-^ -. Cornell University Library QA 371.M98 Introductory course In differential equa 3 1924 004 628 388 The original of tliis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924004628388 INTRODUCTORY COURSE DIFFERENTIAL EQUATIONS INTRODUCTORY COURSE DIFFERENTIAL EQUATIONS FOR STUDENTS IN CLASSICAL AND ENGINEERING COLLEGES BY DANIEL A. MURRAY, B.A., Ph.D. FORMERLY SCHOLAR AND FELLOW OF JOHNS HOPKINS DNIVERSITT INSTRDCTOR IN MATHEMATICS IN CORNELL UNIVERSITY SECOND EDITION NEW YORK LONGMANS, GREEN, AND CO. LONDON AND BOMBAY 1902 COPTEIGHT, 189T, By LONGMANS, GEEEN, AND CO. ALL RIGHTS BBBESTED, First Edition, ipril, 1897. Keprinted July, 1898. Eevised. Eepritited Sept., 1898. July, 1900. September, 1902. T^pogr^pbj bv J 8. Oushing & Co., Norwood, Mass., U. S. A. PEEFACE. The aim of this work is to give a brief exposition of some of the devices employed in solving differential equations. The book presupposes only a knowledge of the fundamental formulae of integration, and may be described as a chapter supplementary to the elementary works on the integral calculus. The needs of two classes of students, with whom the author has been brought into contact in the course of his experience as a teacher, have determined the character of the work. For the sake of students of physics and engineering who wish to use the subject as a tool, and have little time to devote to general theory, the theoretical explanations have been made as brief as is consistent with clearness and sound reasoning, and examples have been worked in full detail in almost every case. Practical applications have also been constantly kept in mind, and two special chapters dealing with geometrical and physi- cal problems have been introduced. The other class for which the book is intended is that of students in the general courses in Arts and Science, who have more time to gratify any interest they may feel in this subject, and some of whom may be intending to proceed to the study of the higher mathematics. For these students, notes have vi PREFACE. been inserted in the latter part of the book. Some of the notes contain the demonstrations of theorems -which are referred to, or partially proved, in the first part of the work. If these discussions were given in full in the latter place, they would probably tend to discourage a beginner. Accordingly, it has been thought better to delay the rigorous proof of several theorems until the student has acquired some degree of familiarity with the working of examples. Throughout the book are many historical and biographical notes, which it is hoped will prove interesting. In order that beginners may have a larger and better conception of the sub- ject, it seemed right to point out to them some of the most important lines of development of the study of differential equations, and notes have been given which have this object in view. For this purpose, also, a few articles have been placed in the body of the text. These articles refer to Riccati's, Bessel's, Legendre's, Laplace's, and Poisson's equa- tions, and the equation of the hypergeometric series, which are forms that properly lie beyond the scope of an introductory" work. In many cases in which points are discussed in the brief manner necessary in a work of this kind, references are given where fuller explanations and further developments may be found. These references are made, whenever possi- ble, to sources easily accessible to an ordinary student, and especially to the standard treatises, in English, of Boole, Forsyth, and Johnson. For students who can afford but a minimum of time for this study, the essential articles of a short course are indicated after the table of contents. PREFACE. vii Of the examples not a few are original, and many are taken from examination papers of leading universities. There is also a large number of examples, which, either by reason of their frequent use in mechanical problems or their excellence as examples per se, are common to all elementary text-books on differential equations. There remains the pleasant duty of making confession of my indebtedness. In preparing this book, I have consulted many works and memoirs ; and, in particular, have derived especial help for the principal part of the work from the treatises of Boole, Forsyth, and Johnson, and from the chapters on Differen- tial Equations in the works of De Morgan, Moigno, Hoiiel, Laurent, Boussinesq, and Mansion. I have in addition to acknowledge suggestions received from Byerly's " Key to the Solution of Differential Equations " published in his Integral Calculus, Osborne's Examples and Rules, and from the trea- tises of Williamson, Edwards, and Stegemann on the Calculus. Use has also been made of notes of a course of lectures deliv- ered by Professor David Hilbert at Gottingen. Suggestions and material for many of the historical and other notes have also been received from the works of Craig, Jordan, Picard, Goursat, Koenigsberger, and Schlesinger on Differential Equa- tions ; from Byerly's Fourier's Series and Sp)lierical Harmonics, Cajori's History of Mathematics, and from the chapters on Hyperbolic Functions, Harmonic Functions, and the History of Modern Mathematics in Merriman and Woodward's Higher Mathematics. The mechanical and physical examples have been obtained from Tait and Steele's Dynamics of a Particle, Ziwet's Mechanics, Thomson and Tait's Natural Philosophy, yiii PBEFACE. Emtage's Mathematical Theory of Electricity and Magnetism, Bedell and Crehore's Alternating Currents, and Bedell's Prin- ciples of the Transformer^ These and many other acknowledg- ments will be found in various parts of the book. To the friends who have encouraged and aided me in this undertaking, I take this opportunity of expressing my grati- tude. And first and especially to Professor James McMahon of Cornell University, whose opinions, advice, and criticisms, kindly and freely given, have been of the greatest service to me. I have also to thank Professors E. Merritt and E. Bedell of the departpient of physics, and Professor Tanner, Mr. Saurel, and Mr. Allen of the department of mathematics at Cornell for valuable aid and suggestions. Professor McMahon and Mr. Allen have also assisted me in revising the proof-sheets while the work was going through the press. To Miss H. S. Poole and Mr. M. Macneill, graduate students at Cornell, I am indebted for the verification of many of the examples. D. A. MURRAY. COKNELL UnIVEKSITY. April, 1897. PREFACE TO THE SECOND EDITION. I TAKE this opportunity of expressing my thanks to my fellow-teachers of mathematics for the kind reception which they have given to this book. My gratitude is especially due to those who. have pointed out errors, made criticisms, or offered suggestions for improving the work. Several of these suggestions have been adopted in preparing this edition. It is hoped that the answers to the examples are now free from mistakes. ^ ^ MURRAY. COBNBLL University, June, 1898. CONTENTS. EQUATIONS INVOLVING TWO VARIABLES. CHAPTER L Definitions. Formation op a Differential Equation. Art. Page l.'tordinary and partial differential equations. Order and degree 1 2.NSolutions and constants of integration 2 S.^The derivation of a differential equation 4 4.SSolutions, general, particular, singular 6 5. Geometrical meaning of a differential equation of the first order and degree 8 6. Geometrical meaning of a differential equation of a degree or an order higher than the first ...... 9 Examples on Chapter 1 11 CHAPTER II. Equations op the First Order and of the First Degree. 8.')liEquations of the form/i(x)da;+/2(j/)«Zj^ =0 . . . .14 9.' (Equations homogeneous in x and y 15 10. ^on-homogeneous equations of the first degree in % and y . 16 ll.'^Exact differential equations 17 12. Condition that an equation of the first order be exact . . 18 IS.SRule for finding the solution of an exact differential equation . 19 14. Integrating factors 21 15. The number of integrating factors is infinite .... 21 16. tintegrating factors found by inspection 22 17. f Rules for finding integrating factors. Rules I. and II. . , 23 18.- Rules III. and IV. 24 ix CONTENTS. Akt. Page 19.^ Rule V 26 20. ^inear equations 26 21. ^Equations reducible to the linear form 28 J^xamples on Chapter II 29 CHAPTER III. EgnATiONS of the Fikst Okder but not op the First' Degree. 22. Equations that can be resolved into component equations of the first degree 31 23. Equations that cannot be resolved into component equations . 32 24. Equations solvable for y 33 25. Equations solvable for x 34 26. Equations that do not contain x ; that do not contain y . .34 27. Equations homogeneous in x and y 35 28. Equations of the first degree in x and y. Clairaut's equation . 36 29. Summary 38 Examples on Chapter III 38 CHAPTER IV. SiNGCLAK Solutions. 30. References to algebra and geometry 40 31. The discriminant 40 32. The envelope 41 33. The singular solution 42 34. Clairaut's equation 44 85. Relations, not solutions, that may appear in the p and c dis- criminant relations 44 36. Equation of the tac-locus 45 37. Equation of the nodal locus 45 38. Equation of the cuspidal locus 47 39. Summary 48 Examples on Chapter IV 49 CHAPTER V. Applications to Geometry, Mechanics, and Physics. 41. Geometrical problems 51 42. Geometrical data 51 CONTENTS. XI Art. Page 43. Examples 44. Problems relating to trajtectories 45. Trajectories, rectangular co-ordinates 46. Orthogonal trajectories, polar co-ordinates 47. Examples 48. Mechanical and physical problems Examples on Chapter V 53 55 55 56 57 58 60 CHAPTER VI. Linear Equations with Constant Coefficients. 49. Linear equations defined. The complementary function, the particular integral, the complete integral . . . .63 50. The linear equation with constant coefBoients and second mem- ber zero 64 51. Case of the auxiliary equation having equal roots ... 65 52. Case of the auxiliary equation having imaginary roots . . 66 53. The symbol Z) ■ . • -67 54. Theorem concerning D 68 55. Another way of finding the solution when the auxiliary equa- tion has repeated roots 69 56. The linear equation with constant coefficients and second mem- ber a function of a; 70 57. The symbolic function 70 58. Methods of finding the particular integral 72 59. Short methods of finding the particular integrals in certain cases 73 60. Integral corresponding to a term of form e'" in the second member 74 61. Integral corresponding to a term of form k™ in the second member 75 62. Integral corresponding to a term of form sin ax or cos ax in the second member 76 63. Integral corresponding to a term of form e" V in the second member '" 64. Integral corresponding to a term of form xF in the second member 79 Examples on Chapter VI 80 xii CONTENTS. CHAPTER VII. Linear Equations with Vakiable Coefficients. Abt. Paoe 65. The homogeneous Ihiear equation. First method of solution . 82 66. Second method of solution: (^) To find the complementary function 84 67. Second method of solution : {B) To find the particular integral 85 68. The symbolic functions /(9) and 86 69. Methods of finding the particular integral . . . . .87 70. Integral corresponding to a term of form a;« in the second member 89 71. Equations reducible to the homogeneous linear form . . 90 Examples on Chapter VII. 91 CHAPTER VIII. Exact Differential Equations and Equations of Particular Forms. Integration in Series. 73. Exact differential equations defined 92 74. Criterion of an exact differential equation 92 75. The integration of an exact equation ; first integrals . . 94 76. Equations of the form ^=/(x) 96 77. Equations of the form -T^ =/(?/) 96 78. Equations that do not contain y directly 97 79. Equations that do not contain x directly 98 80. Equations in which y appears in only two derivatives whose orders differ by two 99 81. Equations in which y appears in only two derivatives whose orders differ by unity 100 82. Integration of linear equations in series 101 83. Equations of Legendre, Bessel, Riccati, and the hypergeometrio series 105 Examples on Chapter VIII 107 CONTENTS. xiii CHAPTER IX. Equations of the Second Okdek. ^^■'^ Page 85. The complete solution in terms of a known integral . . . 109 86. E elation between the integrals IIX 87. To find the solution by inspection Ill 88. The solution found by means of operational factors . . .112 89. The solution found by means of two first integrals . . .114 90. Transformation of the equation by changing the dependent variable 114 91. Removal of the first derivative 115 92. Tran.sformation of the equation by changing the independent variable 117 93. Synopsis of methods of solving equations of the second order '. 118 Examples on Chapter IX 120 CHAPTER X. Geometrical, Mechanicai,, and Physical Applications. 95. Geometrical problems 121 96. Mechanical and physical problems 122 Examples on Chapter X 124 EQUATIONS INVOLVING MORE THAN TWO VARIABLES. CHAPTER XI. Okdinary Differential Equations with siore than Two Variables. 98. Simultaneous differential equations which are linear . . 128 99. Simultaneous equations of the first order .... 130 100. General expression for the integrals of simultaneous equations of the first order 133 101. Geometrical meaning of simultaneous differential equations of the first order and the first degree involving three variables . 134 102. Single differential equations that are integiable. Condition of integrability 136 103. Method of finding the solution of the single integrable equation 137 XIV CONTENTS. Art. Page 104. Geometrical meaning of the single differential equation which is integrable ' . . . 140 105. The locus of Pdx + Qdy + Bdz = is orthogonal to the locus P Q B 106. The single differential equation which is non-integrable . . 142 Examples on Chapter XI. 143 CHAPTER XII. Partial Diffekential Equations. 107. Definitions 146 108. Derivation of a partial differential equation by the elimination of constants 146 109. Derivation of a partial differential equation by the elimination of arbitrary functions 148 Partial Differential Equations of the First Order. 110. The integrals of the non-linear equation : the complete and particular integrals 149 111. The singular integral . . ■ 150 112. The general integral 150 113. The integral of the linear equation „ 153 114. Equation equivalent to the linear equation . . . .154 115. Lagrange's solution of the linear equation .... 154 116. Verification of Lagrange's solution I55 117. The linear equation involving more than two independent variables j5g 118. Geometrical meaning of the linear partial differential equation 158 119. Special methods of solution applicable to certain standard forms. Standard I. : equations of the form F(p, q)=0 120. Standard II. : equations of the form z =px + qy +f(p, q) 121. Standard III. : equations of the form F(z, p, q) = 122. Standard IV. . equations of the form fi{x,p) =fi{y, q) 123. General method of solution 159 161 162 164 166 Partial Differential Equations of the Second and Higher Orders. 124. Partial equations of the second order Igg 125. Examples readily solvable yjn CONTENTS. XV Art. Page 126. General method of solving Iir+Ss+Tt = V. . . .171 127. The general linear partial equation of an order higher than the first 173 128. The homogeneous equation with constant coeflBcients: the complementary function 174 129. Solution when the auxiliary equation has repeated or imagi- . nary roots 175 130. The particular integral 176 181. The non-homogeneous equation with constant coefficients : the complementary function 179 132. The particular integral 180 133. Transformation of equations 182 134. Laplace's equation, V2F=0 182 135. Special cases 185 1.36. Poisson's equation, V''F= — 4 Trp 186 Examples on Chapter XII 187 MISCELLANEOUS NOTES. A. Reduction of equations to a system of simultaneous equations of the first order 189 B. The existence theorem • . 190 C. The number of constants of integration 194 T). Criterion for the independence of constants .... 195 E. Criterion for an exact differential equation . . . .197 P. Criterion for the linear independence of the integrals of a linear equation 197 G. Relations between the integrals and the coefficients of a linear equation 199 H. Criterion of integrability of Pdx + Qdy + Bdz = . . .200 I. Modern theories of differential equations. Invariants . . 202 J. Short list of works on differential equations .... 205 K. The symbol D 208 L. Integration in series 209 Answers to the examples 211 Index of names ......... 233 Index of subjects 235 XVI CONTENTS. SHORT COURSE. (The Roman numerals refer to chapters, the Arabic to articles.) I. ; II. 7-16, 20, 21 ; III. ; IV. 30-34 ; V. ; VI. 49-53, 56-62 ; VII. 65, 66, 71 ; VIII. 72-81 ; IX. 84, 85, 87, 90-93 ; X. ; XI. 97-99, 101- 103, 106; XII. 107-116, llS-122, 124, 125, 127, 128, 131, 133. DIFFEEENTIAL EQUATIONS. CHAPTER I. DEFINITIONS. FORMATION OF A DIFFERENTIAL EQUATION. 1. Ordinary and partial differential equations. Order and degree. A_diff£i:e ntial equation is an equation tha.t involvfts differentials or difFerentia l p.np.ffipip.nts. _ Ordinary differential equations are those in vb ^pli pjl ^^q di ffere ntial coe fficients have reference to a single inde pftnflP"*^ variable. Thus, (1) (2) dy =cosxdx, , . .Ax dy - (y + a) = 0, [ l-(l)' (5) dx are ordinary differential equations. B 1 2 DIFFERENTIAL EQUATIONS. [Ch. I. P aiftial differential eqiuations are those in which the.rp, ax e ^o or more independent variables and partial differential co- effi cients wit h ^ reference to any of them : aSp ,dz , dz dx ay The order nf a difpp.rentia.l eqiiatinn is t.hp. nrdp.r nf t.TiB highest derivative appea.iin^ ip it. The degree of an equation is the degree of .that highest derjj - a tive, when the differe ntial co efficients are free from radicals andjEaciions. Of the examples above/ (1) is of the first order and first degree, (2) is of the second order and first degree, (4) is of the first order and second degree, (6) is of the second order and second degree, (6) is of the first order and" second degree. In the integral calculus a very simple class of differen- tial equations of which (1) is an example have been treated. Equatio ns having one dependent va,ria.l)1fi ?/ and one irtd e- p ej^dent vS lkti £ x will first be considered. The typical form of -such equations is 2. Solutions and constants of integration. Whether a differ - e ntial equation has a solut i on, what are the conditions unde r •ffilUiJl-it-jiU have i\, s olution of a, particular character, and. other qii estions arising in the general theory of the subject a re hardl.Y mat ters for an introductory course.* T ^ student wil l remembCT that he s o lved algebrai c equations, before he coa ld prove_that _such equations must have roots, or before he had iB ttte lilian a , vpry l im ited knowledp-e o£ their p;eneral proper -- tifis. This book will be_ concerned merely with an e xposition ofjhe metho ds of solving" some particuk^dasseToFliHgrMitial ec|uationsj_and their^joluti ons will be expressed by the ordi- Ba,ry algebraic, trigonome^trJc^^aiidexp oliential functions ." * For a proof that a differential equation has an integral, and for references relating to this fundamental theorem, see Note B, p. 190. § 2.] CONSTANTS OF INTEGRATION. 3 A solution or integral of a differential equation is a relation between the variables, by means of which and the derivatives obtain ed therefrom, the equation is sa.tisfied- Thus y = sin a; is a solution of (1) ; ^ x^ + y^ = r^ and y = mx + r Vl + m^ are solut io ns of (4) Art. 1* In two of these solutions, v is expressea explicitly' in terms of -t. hut \-n tVit. ar.1i-|tions of dif- ferential equati ons in general the relg^nn hofwoon ^ QT^r| y \f^ ottentimes not so simply expressed. T his will be seen bv, glancing^at the solutions of the examples on Chapter II. A so luti on o f (1) Art. 1 is y = sin a; ; another solution is y = smx + c,- (1) c being any constant. By ch anging the value of c, differg iiJi. solut ions are obtained, and in particula r, by giving c the value zero, the solution y = smx is obtained. A s olution of d^'^^^ / (^) d'y A/ is y = sin x, and another solution is y = cos x. A solution more eeneral than either o f the former is y = JL sina;; and it includes one of them, as is seen by giving A the particular value unity. Similarlv y = ficosa; includes one of the two first given solu- €i ons of (2). The relation y = A cos x + Bsinx ^ (3) is a yet more general solution, from which all the preceding solu- tions of ('Z\ a.re obtainable by giving particular values to A and B. The arbitrary constants A, B, c, appearing in these solutions are called arbitrary constants of integration.^ Solution _il) has one arbitrary constant, and solution (3) has two ; the question arises : How many arbitrary constants must the most general solution of a diiferential equation contain ? - * See page 12. DIFFERENTIAL EQUATIONS. ICh. 1. T]i&_ answer can mj art be in fer red from the consi deratioiLof the formationof3jlifferentiaijegjiaUiai. 3* The derivation of a differential equation. IjiJiLe-JffScess oppppir Tn p.lvi ninate two constants, A and -B. three equations . aTe_rRf]iiired ^ Of these three equa tio n -, n n n ii r riYeTl, TiFimply , /^v J^^niTlhr'two others needed are_j jl)tfliinprl hy Ruccejjsive diffeiien tiation of - (3) , Thus,-v y = A sin a; + -B cos x, y ^ = ^ cos a; — S sin x, ^ dx ^ ^= — ^sma; — jBcosa;^ dar ■whence, -4 + 2/ = <3<^ d!a;2 Now consider the general process. The equation /(a;, y, c„ Cj, ■ • •, c„) = ^ (1) c ontains, b ^girlpR ■" ""^l i^, « arViH.rg.PY pnTigt.a|^|.g ^^^^- ...,r.^ T>i ffpvBnt.ia.t.i(7r| p, times in succession with respect to x gives fla; % da; d3? ^ 5a; a?/ da; dy'\dxj dy dx\ ' aa!""^ '^dydx" T{x,y,c) = for its general solution, has for its locus a single infinity of curves, there being but one arbitrary constant in (/> ; n of these curves pass through each point of the plane, since -1. has n dx values at any point ; and hence the constant c must appear in the mth degree in the general solution. The general solution of a differential equation of the second order, contains two arbitrary constants, and will therefore have for its locus a double infinity of curves ; that is, a set of curves oo^ in number. Ex.1. ^-0 has for its solution, y = mx + c, m and c being arbitrary. A line through any point (0, c), drawn in any direction m, is the locus of a particular integral of the equation. On taking a particular value of c, say ci, there will be an infinity of lines corresponding to the infinity of values that m can have, and all these lines are loci of integrals. Since to each of the infinity of values that c can have there corresponds an infin- ity of lines, the complete integral will represent a doubly infinite system § 6.] EXAMPLES. 11 of straight lines ; in other words, the locus of that differential equation consists of a doubly infinite system of lines. This can be deduced from other considerations. The condition cPv -^ = requires, and requires only, that the curve described by the moving point shall have zero curvature, that is, it can be any straight line ; and there can be oo^ straight lines drawn on a plane. Ex. 2. All circles of radius r, a^^ in number, are represented by the equation {X - ay + {y - by. = r-', where a and 6, the co-ordinates of the centre, are arbitrary. On elimi- nating a and 6, there appears {■+(8)"}'= '■ (JX2' Thus, the locus of the latter equation of the second order consists of the doubly infinite system of circles of radius r. • Ex. 3. The locus of the differential equation of the third order, derived in Example 1, Art. 3, includes all circles, co' in number; for it is derived from a complete primitive which has a, 6, o arbitrary and thus represents circles whose centres and radii are arbitrary. It will have been observed from the above examples on lines and circles, that as the order of the difi^erential equa- tion rises, its locus assumes a more general character. EXAMPLES ON CHAPTER I. 1. Eliminate the constant a from Vl — x^ + y/l — y^ = a{x — y). 2. Form the differential equation of which y = ce*'"~ "^ is the com- plete integral. 3. Eind the differential equation corresponding to y = ae^ + be-^ + ce', where a, 6, c are arbitrary constants. 4. Form the differential equation of which c(y + cy = x^ is the com plete integral. 6. Eliminate c from y = ex + c — c^. 12 DIFFERENTIAL EQUATIONS. [Ch. I. 6. Eliminate c from ay'^ = (x — c)*. 7. Form the differential equation of which e^ + 2 cae" + c^ = is the complete integral. 8. Eliminate o and h from xy = ae* + be~'. 9. Form the differential equation which has y — a cos (mx + 6) for its complete integral, a and 6 being the arbitrary constants. 10. Form the differential equation that represents all parabolas each of which has a latus rectum 4 a, and whose axes are parallel to the x axis. 11. Find the differential equation of all circles which pass through the origin and whose centres are on the x axis. 12. Form the differential equation of all parabolas "whose axes are parallel to the axis of y. 13. Form the differential equation of all conies whose axes coincide with the axes of co-ordinates. 14. Eliminate the constants from y = ax -\- bx^. Note. [The following is intended to follow line 6, page 3.] Ex. 1. Show that x^ + y^ = r^ is a solution of equation (4) Art. 1. Differentiation gives x + y -^ = 0, whence' -^ = ux cix y du x^ \ ^ Substitution of this value of -^ in 4 gives y = \- j-v/l + — , dx y ^ y^ which reduces to x^ + y^ = r'^. Ex. 2. Show that y Differentiation gives Ex. 2. Show that y ~mx -^ rVl + ir? is a solution of (4) Art. 1. <3.x~ dy Substitution of this value of -=^ in (4) gives y = mx + rvT+m^. Ex. 3. Show that x^ + iy = is a solution of [^] + x-^ -y = 0. Ex. 4. Show that y = ax'' + hx is a solution of^--^ + ^ = 0. " dx;' X dx x^ Ex. 5. Show that v = — +B is a solution of — + - — = 0. )• dr^ r dr Ex. 6. Show that y = ae'" + be-'^ is a solution of y| - k''y = 0. § 7.] FIBST ORDER AND FIRST DEGREE. 13 CHAPTER II. EQUATIONS OF THE FIRST ORDER AND OF THE FIRST DEGREE. 7. In Chapter I. it has been shown how to deduce from a given relation between x, y, and constants, a relation between X, y, and the derivatives of y with respect to x. There has now to be considered the inverse problem : viz., from a given relation between x, y, and the derivatives of y, to find a re- lation between the variables themselves. As, for instance, the problem of finding the roots of an algebraic equation is more difficult than that of forming the equation when the roots are given; or as, again, integration is a more difficult process than differentiation-, so here, as in other inverse proc- esses, the process of solving a differential equation is much more complicated and laborious than the direct operation of forming the equation when the general solution is given. An equation is said to be solved, when its solution has been reduced to expressions of the forms if{x)dx, \ {y)dy, even if it be impossible to evaluate these integrals in terms of known functions. The equation f(^>y>%-->-^nj = ^ ^'^^^^^ ^ ^"^^^"^ ^"^ every case. In fact, even P-^ -f- Q = 0, where P and Q are functions of x and y, cannot be solved completely. It will be remembered how few in number are the solvable cases in algebraic equations ; and it is the same with differential equa- tions. The remainder of this book will be taken up with a 14 DIFFERENTIAL EQUATIONS. [Ch. II. consideration of a few fecial forms of equations and the methods devised for their solution.* This chapter will be devoted to certain kinds of equations of the first order and degree, viz. : 1. Those that are either of the form f{x) dx +fi{y) dy = 0, or are easily reducible to this form ; 2. Those that are reducible to this form by the use of special devices : — (a) Equations homogeneous in x and y. (b) Non-homogeneous , equations of the first degree in x and y; 3. Exact differential equations, and those that can be made exact by the use of integrating factors ; 4. Linear equations and equations that are reducible to the linear form. 8. Equations of the form fi(x)dx+f2(y)dy — 0. When an equation is in the form fix)dx+f,(y)dy = 0, its solution, obtainable at once by integration, is jf (x) dx + J/j (y) dy = c. If the equation is not in the above form, sometimes one can see at a glance how to put it in that form, or, as it is com- monly expressed, to separate the variables. Ex. 1. (!■) (1 - x)d!/ - (1 + y)dx = can evidently be written (2) -^ ^ = 0, 1-i-y 1-x ' * Tiie student who is proceeding to find tlie methods of solving dif- ferential equations with no more knowledge of the subject than that imparted In the preceding pages, is reminded that he does this, assuming (1) that every differential equation with one independent variable has a solution, and (2) that this solution contains a number of arbitrary con- stants equal to the number indicating its order. §8,9.] HOMOGENEOUS EQUATIONS. 15 whence, on integrating, (3) log(l + ?/)+log(l-a;)=c, and hence (4) (l + 2,)(l-a;)=e<= = ci. In equations (3) and (4) appear two ways of expressing a general solution of the same' equation. Both are equally correct and equally general, but the one has the advantage over the other in neatness and simplicity, and this would make it more serviceable in applications. In some of the examples set, the reduction of the solutions to forms neater and simpler than those which at first present themselves, may require as much labour as the solving of the equations. The solution (4) could have been obtained without separating the variables, if one had noticed that (1 — x)dy —(14- y)dx is the diiierential of (1 — x)(l + y). Here, as in the calculus and other subjects, the experience that comes from practice, is the best teacher for showing how to work in the easiest way. Equa- tion (1) can also be put in the form dy — dx — {xdy + y dx) = 0, and another form of the solution obtained, namely, y-x-xy = C2. Solution (4) reduces to this form on putting d for ci — 1. Ex.3, solve (,_.!) = a(,^ + 1). Ex.4. Solve Se'^ta,nydx+{1 —e')sec^ydy = 0. < 9. Equations homogeneous in x and y- These equations can be put in the form dx fs,(x,y)' where /i, /a, are expressions homogeneous and of the same degree in x and y. On putting y=vx, this equation becomes ax 16 DIFFERENTIAL EQUATIONS. [Ch. II. since each term in/i, /a, is of the same degree, say n, in a;; and x" is thus a factor common to both numerator and denominator of its right-hand member. Separation of the variables gives dv _dx F{v) —v~x' the solution of which gives the relation between x and v, that is, between y and ^, which satisfies the original equation. Ex. 1. Solve (^x'' + y'^)dx-2xydy = 0. Putting y = vx gives (1 + v^)dx -2v(^xdv + vdx)=0, which, on sepa- ration of the variables, reduces to dx 2v ^ n X 1-v^ Integrating, log a;(l - v^) = log c. On changing the logarithmic form to the exponential, and putting ^ for V, the .solution becomes * •a;2 — 2/2 = ex. Ex. 2. Solve j/2 dx + (xy + x^)dy = 0. Ex. 3. Solve x^ydx - (a;' -|- y^)dy = 0. Ex. 4. Solve (iy + 3x)^+y-2x = 0. dx « 10. Non-homogeneous equations of the first degree in x and y. These equations are of the form dy ^ ax+by + G _ m\ dx a'x + h'y + c'" ' For X put x' + h, and for y put y' + k, where h and k are constants ; then dx = dx' and dy = dy', and (1) becomes dy' _ ax' + by' + ah + bk + c dx' " a'x' + b'y' -\- a'h + b'k + c'' If h and k are determined, so that a/i + 6fc -f- c = 0, § 10, n.] EXACT DIFFERENTIAL EQUATIONS. 17 and . a*h + b'k + c' = 0, then (1) becomes ^ = 7' + ^f„ (2) which is homogeneous in x' and y\ and therefore solvable by the method of Art. 9. If (2) has for its solution /(»=', 2/0=0. the solution of (1) is fix — h), (y — k)= 0. This method fails when a: b = a' : b', h and k then being infinite or indeterminate. Suppose a _b__l^ a' V m then (1) can be written dy _ ax + by + c dx m(ax + by) + c' On putting v for ax + by, the latter equation becomes dv , , V + G — = a + b ■ — J dx mv + c where the variables can be separated. Ex.1. Solve (3y-Tx + 1)dx+(,7y-Zx+3)dy = 0. Ex.2. Solve (y.-Sx + 3)^ = 2y-x-i. 11. Exact differential equations. A differential equation which has been formed from its primitive by differentiation, and with- out any further operation of elimination or reduction, is said to be exact; or, in other words, an exdct differential equation is formed by equating an exact differential to zero. There has now to be found the condition which the coefficients of an equar tion must satisfy, in order that it may be exact, and also the method of solution to be employed when that condition is DIFFERENTIAL EQUATIONS. j fClH.-II 18 satisfied. The question of how to proceed when the | condition is not satisfied will be considered next in order. 12. Condition that an equation of the first order be exkj; What is the condition that Mdx + Ndy = (1) be an exact differential equation, that is, that Mdx + Ndy be an exact differential ? In order that Mdx + Ndy be an exact differential, it must have been derived by differentiating some function m of a; and y, and performing no other operation. That is, du = Mdx + Ndy. But du = ^dx+^dy. ax dy Hence, the conditions necessary, that Mdx + Ndy be the differ- ential of a function m, are that ilf=^\ and J\r=^. (2) dx dy ^ ' The elimination of m imposes on M, N, a single condition, ^=m (3) dy dx d'u since each of these derivatives is equal to - — —• dx dy This condition is also sufficient for the existence of a func- tion that satisfies (1).* If there is a function u, whose differ- ential du is such that -J du = Mdx -f- Ndy, then on integrating relatively to x, since the partial differential ' Mdx can have been derived only from the terms containing x, u= I Mdx + terms not containing x, that is, u = CMdx + F{y). (4) * For another proof see Note E. § 12, 13.] EXACT DIFFERENTIAL EQUATIONS. 19 Differentiating both sides of (4) with respect to y, dy dyj dy But by (2), -— must equal N, hence dy i^M^N-±CMdx. . (5) dy dy-J The first member of (5) is independent of x ; so, also, is the second : for differentiating it with respect to a; gives , dx dy which, by condition (3), is zero. Integration of both sides of (5) with respect to y gives F(y) =f{N- ^JMdx ^dy + a, where a is the arbitrary constant of integration. Substitution in (4) gives u = C Mdx+ C \ N-j- C Mdx Xdy + a. Therefore the primitive of (1), when condition (3) is satis- fied, is CMdx + C\n- ^C Mdx \dy = c. (6) Similarly, C Ndy +J^\^-yS ^'^^ [ '^^^ = " is also a solution. 13. Rule for finding the solution of an exact differential equation. Since all the terms of the solutiou that contain x must appear in CMdx, the differential of this integral with respect to y must have all the terms of Ndy that contain x; and therefore (6) can be expressed by the following rule : To find the solution of an exact differential equation, Mdx + Ndy = 0, integrate Mdx as if y were constant, integrate the terms in Ndy that do not give terms already obtained, and equate the sum of these integrals to a constant. 20 DIFFERENTIAL EQUATIONS. [Ch. II. Ex. 1. Solve (^x'^-ixy -2y'^)dx + (y' -ixy -2x,^)dy = 0. Here, S. — = _4a;_4j/ = S£L ■ hence it is an exact equation. dy dx JMdx is 2 x^ — 2 xy^ ; y'^dy is the only term in Ndy free from x. o Therefore the solution is ^-2x^y-2xy^ + '^=ci, o 6 or x^ — 6 x^y — 6 xy^ + j/S = c. The application o£ the test and of the rule can sometimes be simplified. By picking out the terms of Mdx + Ndy that obviously form an exact differential, or by observing whether any of the terms can take the form f{v.)dn, an expression less cumbersome than the original remains to be tested and integrated. For instance, the terms of the equation in this example can be rear- ranged thus : x'^dx + y'^dy -{ixy + 2y^)dx-{ixy + 2x'^)dy = 0. The first two terms are exact differentials, and the test has to be applied to the last two only. ,Ex.2. xdx + ydy-,^-^:^ = ^ x^ + y' becomes, on dividing the numerator and denominator of the last term bya;2, xdx + ydy + — i— ^ = 0, ■ '+© each term of which is an exact differential. Integrating, — 5-^ + tan-i^ = c. 2 X Ex. 3. Solve ((12 -2xy- y^)dx -(x + yydy = 0. Ex. 4. Solve (2 ax-\-hy+ g)dx + (2 cy + 6x + e)dy = 0. Ex. 5. Solve (2 x'^y + ix^ - I2xy''' + 3y^ - X& + e'^)dy + (12a;2y+ 2a;!/2 + \3? - 4y« + 2ye^ - e»)da; = 0. / § 14, 15.] INTEGRATING FACTORS. 21 14. Integrating factors. The differential equation ydx — xdy = is not exact, but when multiplied by — , it becomes y ydx — xdy _(^ which is exact, and has for its solution X _ y~ When multiplied by — , the above equation becomes xy dx dy . ^ = 0, X y which is exact, and has for its solution log x-\ogy = c, which is transformable into the solution first found. Another factor that can be used with like effect on the same equation . 1 Any factor a, such as —, — , — „, used above, which changes y' xy or an eqj-iation into an exact differential equation, is called an integr^H/Kmctor. 15. The number of integrating factors is infinite. The num- ber of integrating factors for an equation Mdx + Ndy = 0, is infinite. For suppose /a is an integrating factor, then /x (Mdx + Ndy) = du, and thus ?< = c is a solution. Multiplication of both sides by any function of ?t, say/(M), giyes fjif(ti){Mdx + Ndy) =f(u)du ; 22 DIFFERENTIAL EQUATIONS. [Ch. II. but the second member of the last equaltion is an exact differ- ential ; therefore the first is also, and hence ju./(m) is an inte- grating factor of the equation Mdx + Ndy = ; and as /(m) is an arbitrary function of n, the number of in- tegrating factors is infinite. This fact' is, however, of no special assistance in solving the equation. 16. Integrating factors found by inspection. Sometimes inte- grating factors can be seen at a glance, as in the example of Art. 14. Ex. 1. Solve ydx — xdy + log xdx = 0. Here log xdx is an exact differential; and a factor is needed for ydx — xdy. Obviously"— is the factor to be employed, as it will not affect the third term injuriously, from the point of view of integration. The exact equation is then ydx -xdy , loga^^ _ „ ^2 +-^ «»;-". the solution of which reduces to CX + y + \ogx + 1 =0. ) Ex. 2. Solve (1 -I- xy)ydx + {I - xy)xdy = 0. Rearranging the terms, ydx + xdy + xy^dx - xh/ dy = 0, -\ that is, d{xy} + xy'^dx - xhj dy = 0. J' For this, the factor -^ immediately suggests itself, and the equation X y becomes t^C^?/) dx dy ^■'y'^ X y ~ ' Integrating, -'■- + log- = ci. xy °y 1 and transforming, It will be well to try to iind an integrating factor by inspection, before having recourse to the rules given in Arts. 17, 18, 19. § 16, 17.] BVLES FOR INTEGRATING FACTORS. 23 Ex. 3. a(xdy + 21/ dx) = xy dy. Ex. 4. (x*e» - 2 mxy'^) dx + 2 mx^y dy = Q. — Ex. 5. y(2xy + e') dx - e'dy = 0. 17. Rules for finding integrating factors. Rules I. and II. Kules for finding integrating factors in a few cases will now be given.* Rule I. When Mx + Ny is not equal to zero, and the equa- tion is homogeneous, — -— is an integrating factor of MX + Ny ° Mdx + Ndy = 0. Rule II. When Mx — Ny is not equal to zero, and the equation has the form fi (xy) ydx+fs (xy) xdy = 0, -— — - is an integrating factor. Mx — Ny Proof : Mdx + Ndy = l[(,Mx+Ny)(^ + ^\ + (Mx-Ny)(^-^\j is an identity. Tliis may be written, (a) Mdx + Ndy = J | (Mx + Ny)d- log xy + (Mx - Ny)d-los^\- Division of (a) by Mx + Ny gives Mdx + Ndy , , , , , Mx - Ny _, , x — jT — , ,,. " = id-logxy + l^Tf ^d-log- Mx + Ny ^ '' " ^ Mx + Ny ^y Now if Mdx + Ndy is a homogeneous expression, -^ ^ is homo- ^ " ' Mx + Ny geneous and equal to a function of -, and Mdx + Ndy , , , , ^fx\ , , x -^—^=id.losxy + if[-)d.los-, or, smce - = e'os;-, y ' * For a discussion on and determination of integrating factors, see George Boole, Differential Equations, pp. 55-90. 24 DIFFERENTIAL EQUATIONS. [Ch. II. ±^=ia.losxy + iF[log^)d.log'^, Mdx + Ndy which Is an exact differential. On dividing (a) by Mx — Ny, it becomes, Mdx + Ndy ,Mx+Ny,. , , , , x -nr JF-^ = i TTr i7^ (i • log a;!/ + i d . log -, Mx — Ny Mx — Ny . ^ » ' 2 & ^^ and if Mdx + iVS^ is of the ioxm. fi(xy)y dx +f2(_xy)xdy, this will be Mdx + Ndy , fi{xy)xy + fi{xy)xy , , . ,, , x M^-Ny =^ Mxy)xy-Mxy)xy ^-"'^^y+'^^-''^' = Fi(xy)d -logxy +ld- log-, = Fi{\ogxy)d ■ logxy + J i • log-, which is an exact differential. "When Mx+ Ny = 0, — = - ^. Substitution for — in N X N Mdx + Ndy = and integration gives the solution x = ey. When Mx- Ny = 0, — = 1^. Substitution for — in the differential N X N equation and integration gives the solution xy = c. Ex. 1. Solve {niy - 2 xy^)dx - (»» - Zx^y)dy = 0. Ex. 2. Solve Ex. 3, Art. 9, by this method. Ex. 3. Solve y(xy + 2 x^)dx + x{xy - xhp')dy - 0. 18. Rules III. and IV. EuLB III. When ^ , is a function of x alone, say /(«), gj/(i)& ig an integrating factor. § 18, 19.] RULES FOB INTEGRATING FACTORS. 25 For, multiplication of Mdx + Ndy = by that factor gives, say, Midx + Njdy = ; and differentiation will show that dM, ^ dNi dy dx Ex. 1. (a;2 + y^ + 2x)dx + 2 ydy = 0. Ex. 2. (x2 + y^)dx -2xydy = 0. i " ■ dN dM EuLE IV. When — ^ is a function of y alone, say F(j)), gSKy)ds is an integrating factor. This can be shown in the same way as in the preceding rule. Ex. 3. Solve (3 a;V + 2 xy)dx + (2 a^j/S - x'^}dy = 0. Ex. 4. Solve (y* + 2y)dx+ (xy^ + 2 j/* - 4 x)dy = 0. 19.* Rule V. a;'"""'""?/''""'"^, where k has any value, is an integrating factor of x^y^ {my dx + nxdy) = 0, for on using the factor, the equation becomes -d(a;''"2/"")=0. K Moreover, when an equation can be put in the form x'^^(mydx + nxdy) + x'^y^^{mjydx + nixdy)=0, an integrating factor can be easily obtained. A factor that will make xfy^{viydx + nxdy) an exact differential is a;'"'-'-»?/''»-'-P, where k has any vahie ; and a factor that will make af^y^i (wii?/ dx + n^x dy) an exact differential is a!''i"i-i-°i2/''i"i-'-Pi, where kj has any value. * See L'Abb6 Moigno, Calciil Differentiel et Integral (published 1844), t. II., Nb. 147, p. 355; Johnson, Differential Equations, Art. 32. 26 DIFFERENTIAL EQUATIONS. [Ch. II. These two factors are identical if Km — 1 — « = KiWii — 1 — «i, and KW — 1 — /? = KiUi — 1 — /8i. Values of « and k^ can be found to satisfy these conditions. Ex. 1. Solve (!/8 - 2 yx^)dx + (2 xy'^ - x^)dy = 0. Rearranging in the form above, 2/2(2/ dx + 2x dy) — x^(2ydx + x dy) = 0. For the first term a = 0, /8 = 2, m = 1, re = 2, and hence x«-'2/^''~^~^ is its integrating factor. For the second term a = 2, ;3 = 0, m = 2, re = 1, and hence x^i^'-'^'-h/"'-^ is its integrating factor. These factors are the same if k-1=2k'-1-2, and 2k-1-2 = k'-1. On solving for « and k', k = 2 = k', and therefore xy is the common integrating factor for both terms. The equation when made exact is xy {y^(y dx + 2xdy)- x'^{2ydx + x dy")} ^ 0. ^ - ^ = c, or a;V(2/2 - x^) = o. Ex.2. Solve (2 x^y - 3y*)dx + (3x> + 2 xy'') dy = 0. Ex. 3. Solve iy^ + 2x^y')dx + (2x^-xy)dy = 0. 4 20. Linear equations. A differential equation is said to be linear when the dependent variable and its derivatives appear only in the first degree. The form of the linear equation of the first order is where P and Q are functions of x or constants. The solution of ^ + p„ = o, that is, of ^ = ~Pdx, y "§20.] LINEAR EQUATIONS. 27 is y = ce'!'"'^, or yei'"'^ = c. On differentiation the latter form gives ei^^(dy + Pydx) = Q , which shows that eJ^'^ is an integrating factor of (1). Multiplication of (1) by that factor changes it into the exact equation, ei^'^{dy + Pydx) = eS''^Qdx, which on integration gives 2/eJ^*== CeS^'^Qdx + c, or y = e-!''^ j CeS'''^Qdx + c I • (2) The latter can be used as a formula for obtaining the value of 2/ in a linear equation of the form (1).* The student fs advised to make himself familiar with the linear equation and its solution, since it appears very frequently. Ex. 1. Solve x^ — ay = x+ 1. dx This is linear since it is of the first degree in y and ^- Putting it in the regular form, it becomes dy a _x + l dx X X Here P = — -, and the integrating factor e' is — X x' Using that factor, the equation changes to Lcly-^dx=^dx. x' J X''+^ whence y = -^ 1- cx". I — a a * Gottfried Wilhelm Leibniz (1646-1716), who, it is generally admitted, invented 'the differential calculus independently of Newton, appears to have been the first who obtained the solution (2). 28 DIFFERENTIAL EQUATIONS. [Ch. IF. The values of P and Q might have been substituted in the value of y as expressed in (2). Ex. 2. Solve ?^+y = e-'. Ex. 3. Solve oos2 a; ^ + « = tan a:. dz " Ex. 4. Solve (x + 1)^ - nw = e»(x + l)»+i. da Ex. 5. Solve {x^ + l)^ + 2xij = i x"^. 21. Equations reducible to the linear form. Sometimes eqiia^- tions not linear can be reduced to the linear form. In particu- lar, this is the case with those of the form* %+Py=Qy'', (1) where P and Q are functions of x. For, on dividing by ■tf and multiplying by (— w + 1), this equation becomes {-n + 1)2/-" ^ + (- « + 1) Py-""*' = (- n + 1) Q; on putting v for y"""^', it reduces to ^ + {l-n)Pv = (l-n)q, which is linear in v. Ex. 1. Solve $^ + i w = x'^y^. dx x" " Division by j/" gives y-^ ^ +■— = x^ dx X On putting v for y-^, this reduces to -;; v = — 5xK the linear form. dx X Its solution is v = y-^ = cx^ + ■ * This is also called Bernoulli's equation, after James Bernoulli (1654- 1705), who studied it in 1695. §21.] EXAMPLES. 29 Note . In general, an equation of the form where Pand Q are funotiofis of x, on the substitution of v for/(y) becomes ax which is linear. Ex. 2. Solve (1 + y^)dx = {tB,n~^ y - x)dy. This can be put in the form dx 1 tan-' y dy 1 + 2/2-1 + ^2' which is a linear equation, y being taken as the independent variable. Integration as in the last article gives the solution x = tan-i 2/ - l-f ■ce-'^^'^y. Ex. 3. Solve '^ + -y = 3xV- dx x" " Ex. 4. Solve ^+_^ = a:j/^. dx \ — x^ Ex. 5. Solve3a;(l -x2)j/2^ + (2x2 -1)2/3 = 0x3. EXAMPLES ON CHAPTER II. Equations can sometimes he reduced to standard forms by substitutions. l.(a;+2/)2^=a2. \¥\xtx+y=v.-\ 5.{3? -yx'^)'^+y'^ + xy^ = -xy. 19. y + 2/cosa: = !/»sin2a;. 22. ydx + Cax^y -2x)dy = 0. 20. (x+ 1)^+1= 2e-». 23. (1 + 6^2 - 3a:2?/)^ = Sxj^a-a;^. ax "X 24. j/(x2 + ^2 + a2)^ + x(x2 + j/2 - o2) = 0. 25. (x2y3 + X2/)(J2/ = dx. 29. y dy + by'^dx = a cos x dx. V. -„dy dy S'^=«^- 30. 2xydx+(,y^-x^)dy = 0. 27. V^M^__^2, = Va2 + x2-x. 31. (xy^ ^ e'^)dx ~ x"-y dy = 0. 28. (x + ,)| + (x-,) = 0. 32. ,-x| = .(n-x2|). 33. (3j/ + 2x + 4)dx-(4x+62^ + 5)rf!/ = 0. 34. (x'y» + x^y^ + xy + l}y + (x^yS _ x'^y^ ~ xy + l)x^ = 0. 35. (2x'^y''+y)dx-(x»y-3x)dy=0. 37. ^ + !?« = £. ax X X" § 22.] EQUATIONS NOT OF THE FIRST DEGREE. 31 CHAPTER III. EQUATIONS OF THE FIRST ORDER, BUT NOT OF THE FIRST DEGREE. 22. Equations that can be resolved into component equations dy of the first degree. In what follows, ~ will be denoted by p. The type of the equation of the first order and nth degree is p" + P.p"-'- + P^p"-^ + • • • + Pn-lP + i^n = 0, (1) where P^, Pj, •••, P„, are functions of x and y. Two cases appear for consideration, viz. : (a) where the first member of (1) can be resolved into rational factors of the first degree ; (b) where that member cannot be thus factored. In the first case (1) can take the form {p-R,){p-R,)-ip-R:) = (i. (2) Equation (1) is satisfied by a value of y that will make any factor of the first member of (2) equal to zero. Therefore, to obtain the solutions of (1), equate each of the factors in (2) to zero, and obtain the solutions of the n equations thus formed. The n solutions can be left distinct or combined into one. Suppose the solutions derived for (2) are /i {x, y, Ci) = 0, /2 (x, y,c,) = 0,--, /„ (x, y, c„) = 0, where Cj, c^, ■■-, c„, are the arbitrary constants of integration. These solutions are evidently just as general, if Ci = Cj = ••• = c„, since all the c's can have any one of an infinite number of values ; and the solutions will then be 32 DIFFERENTIAL EQUATIONS. [Ch. III. /i («, y, c) = 0, f^{x, y, c) = 0, ••-,/„ (x, y, c) = 0. These can be combined into one equation ; namely, /i (a^) y, c)/2 (x, y,c).--f„ (x, y, c) = 0. (3) Ex. 1. p^ + 2xp^ - y^p'' — 2 xy^p = 0, can be written p(p + 2x)(^p-y^) = 0. Its component equations are p = 0, p + 2x = 0, p~y^ = 0, of which the solutions are y = e, y + x^ = c, and xy + cy + l=0, respectively. The combined solution is (2/ - c)(j/ + a;2 - e)(xy + cy + 1) = 0. When the equation in p is of the second degree, sometimes the solution readily presents itself in the form (3) as in the next example. Ex.2. Solve (^^J-ax'^ = 0. dx Integrating, j, -I- c = ± f a%i Rationalizing, 25(y + c)^ = 4 ax^, or 25(y + c)^-iax^ = 0. > Ex. 3. Solve p\x^2y)\ Zp\x + j^) + (j/ + 2 a;)p = 0. Ex.4. Solve f^y=aa;*. Ex. 5. Solve 42/2p2 + 2jjxy(3 a; + 1) + Ssc^ = 0. ■ Ex. 6. Solve i>2 _ 7^ + 12 = 0. 23. Equations that cannot be resolved into component equations. Methods which may be tried for solving equation (1) of the last article, when its first member cannot be resolved into rational linear factors, (case (6) Art. 22), will now be shown. That equation, which may be expressed in the form /(ar,2/,p) = 0, may have one or more of the following properties. §23,24.] EQUATIONS SOLVABLE FOR y. 33 (a) It may be solvable for y. (b) It may be solvable for x. The case where it is solvable for p has been considered in the preceding section. (c) It either may not contain x, or it may not contain y. (d) It may be homogeneous in x and y. (e) It may be of the first degree in x and y. 24. Equations solvable for y. When the condition (a) holds, f{x, y, p) =0 can be put in the form y = F(x,p). Differentiation with respect to x gives dp\ which is an equation in two variables x and p ; from this it may be possible to deduce a relation "/- (a;, P, c) = 0. The elimination of p between the latter and the original equation gives a relation involving x, y, and c, which is the solution required. When the elimination of p between these equations is not easily practicable, the values of x and y in terms of j3 as a parameter can be found, and these together will constitute the solution. Ex. 1. Solve X — yp = a^. X- ap^ Here y = • Differentiating and clearing of fractions, This can be put in tlie linear form ^ P -f{p) where the variables are separated. Ex. 1. Solve 2/2 _|_ xyp - x^p^ = 0. Ex. 2. Solve y = yp^ + 2px. 28. Equations of the first degree in x and y. Clairaut's equation. When the condition (e) Art. 23, holds, the equation, being solvable for x, and for y as well, comes under cases (a) and (b) considered in Arts. 24, 25. However, there is one particular form of these equations of the first degree in x and y that is of special importance, namely, y=px+f{p), which is known as Clairaut's equation* Differentiation with respect to x gives P=P + l^+f'ip)\% whence a; + /' (p) = 0, ^ = 0. da; * Alexis Claude Clairaut (1713-1765), celebrated for his researches on the figure of the earth and on the motions of the moon, was the first who had the idea of aiding the integration of differential equations by differentiating them. He applied it to the equation that now bears his name, and published the method in 1734. §28.] EQUATIONS OF FIRST BEGBFE IN x AND y. 37 Prom the latter equation, it follows that i? = c, and hence y = cx+f{c) is the solution. The equation a; +/' (p) = is considered in Art. 34. Any equation satisfying condition (e) can be put in the form y = ^fi{p) + Mp)- If /,(p) =p, it is in Clairaut's form. By proceeding as in Art. 24 and differentiating with respect to x there is obtained P=f.{p) + Wi'i.p)+f^'{p)\% . dx_ f,'{p) ^ f,'(p) ■ ■ dp p-Mp) p-Mp)' which is linear in x ; and from this a relation between x and p may be deduced. The student should be familiar enough with Clairaut's form to recognize it readily. Some equations are reducible to this form; Ex. 2 is an illustration. Ex.1. Solve ij = (l +p)x+p^. Differentiating, P = 1 + i) + (» + 2p) ^• .-. ^ + x=-2p, which is linear. Solving, and hence y = 1-p^^^(\^-p) ce-J" from the given equation. Ex. 2. Solve i?{y — px) = yp^. On putting x^ = u, and y- = v, the equation becomes dv /dvy ~ du \du/ ' which is Clairaut's form, . • . V = cu + c^, and hence V^ = cx^ + c^. 38 DIFFERENTIAL EQUATIONS. [Ch. III. Ex. 3. Solve y = xp + sin -^p. Ex. 4. Solve e*'(p - 1) + e^ip" = 0. Ex. 5. Solve xy(y —px) = x +py. Solving foi' X ov y may be of service in the case of equations of the first degree inp ; this is illustrated in Ex. 6. Ex. 6. Solve -^ + 2 xw = a;-^ + ««. ax The solution for y gives the equation j/ = x + Vp, which is of the form discussed in Art. 24. The solution is v = x + '^ '^ " . 29. Summary. What has been said in this chapter con- cerning the equation /(x, y, p) = 0, of degree higher than the first in p, may be thus summed up : Either solve f{x, y, p) = for p, and obtain a solution cor- responding to each value of p ; or, Solve for y or x, and, by differentiating with respect to x or y, obtain an equation, whence another relation between p and X or y can be found. This new relation, taken in connection with the original equation, will serve either for the elimina- tion of p, or for the evaluation of x and y in terms of p ; the eliminant or the values of x and y will be the solution. EXAMPLES ON CHAPTER III. v2. y=^(x-6)+«. 6. ayp''+{2x-b)p-y = 0. 3. xy^p^ + 2) = 2P2/3 + ^s. T y-px=^/l+ p^,p(x^ + y^). J5. p=_9p + i8 = o. 9- i^p-yy=p'-^lp + -i- 10. 3pY -2xyp + iy^-x'^ = 0. 11. (a;2 + 2,2) (1 + p)2 _ 2 (X + 2/)(l + p) (x + yp) + (x -|- 2,p)2 = 0. § 29.] EXAMPLES. 39 13. 2,Hi-p^)=6- ' ^ <^'-^y\ ^~^l P 14. {px-y)(,py + x)= h^p^ IT. x + -^y== = «■ 15. jfl + 2py cot X = y^. 18. ?/ — 2px =f{xp^). 19. a;yp2 + p{Sx^ - 2y^)- 6xy = 0. 20. p8 _ 4 a;jip + 8 2/2 = 0. 21. p3 _ (a;2 + K!/ + 2/2)p2 + (a;3y + a;2i/2 + xj/S)^ - ay = 0. ./22. p« + mp'' = aiy + mx). ^5. 2, _(i +p2)-J = 6. 23. eS'Cp - 1) + p^e^ = 0. m ^ ^ ^ 26. y=px + — 40 DIFFERENTIAL EQUATIONS. [Ch. IV. CHAPTER IV. SINGULAR SOLUTIONS. 30. References to algebra and geometry. In this explanation of singular solutions,* use will be made of a few definitions and principles of algebra and geometry; particularly of the discriminant in the one, and of envelopes in the other. Arti- cles 31 and 32 will serve to recall some of them. The student is advised to consult a work on the theory of equations and a differential calculus concerning these points. 31. The discriminant. The discriminant of an equation in- volving a single variable is the simplest function of the coeffi- cients in a rational integral form, whose vanishing is the condition that the equation have two equal roots. For exam- Ty _|_ -\/h^ 4- etc pie, the value of x in ax' + 6a; -f- c = is =^-- ; and so ^ CL the condition that the equation have equal roots is that &^ — 4 ac be equal to zero. The discriminant is &^ — 4ac; the equation 6- — 4ac = will be called the discriminant relation. * Leibniz in 1694 (see footnote, p. 27), Brook Taylor (1085-1731), the discoverer of the theorem called by his name, in 1715, and Clairaut (see footnote, p. 36) were the first to detect singular solutions of differential_ equations. Clairaut refers to these solutions in a paper published in the Memoirs of the Paris Academy of Sciences in 1734. Their geometrical significance was first pointed out by Lagrange (see footnote, p. 155) in an article published in the Memoirs nf the Berlin Academy of Sciences in 1774, in which he also showed a way of obtaining them. The theoiy at present accepted is that expounded by Arthur Cayley (1821-1895) in an article in the Messenger of Mathematics, Vol. II., 1872. § 30-32.] THE ENVELOPE. 41 When the equation is quadratic, the discriminant can be written immediately ; but when it is such that the condition for equal roots is not easily perceived, the discriminant is found in the following way. The given equation being F=Q, form another equation by diif erentiating F with respect to the vari- able, and eliminate the variable between the two equations. For example, ^{x,y, c) = may be looked on as an equation in c, its coefficients then being functions of x and y. The simplest rational function of x and y, whose vanishing expresses that the equation <^ (x, y, c) = has equal roots for c, is called the c discriminant of <^, and is obtained by eliminating c between the equations. Thus the c discriminant relation represents the locus, for each point of which <^ (x, y, c) = has equal values of c. Similarly, the p discriminant oif{x, y,p) = 0, the differential equation corresponding to (x, y, c) = 0, c be given all possible values, there is obtained a set of curves, infinite in number, of the same kind. Suppose that the c's are arranged in order of magnitude, the successive c's thus differing by infinitesimal amounts, and that all these curves are drawn. Curves corre- 42 DIFFERENTIAL EQUATIONS. [Ch. IV. spending to two consecutive values of c are called consecutive curves, and their intersection is called an ultimate point oj intersection. The limiting position of these points of intersec- tion includes the envelope of the system of curves. It is shown in works on the differential calculus, that the envelope is part of the locus of the equation obtained by eliminating c between '^{x,y, c) = 0, and ^ - • that is, the envelope is part of the locus of the c discriminant relation. This might have been anticipated, because in the limit the c's for two consecutive curves become equal, and the c discriminant relation represents the locus of points for which <^(^) 2/j ") = will have equal values of c. It is also shown in the differential calculus, that at any point on the envelope, the latter is touched by some curve of the system; that is, that the euvelope and some one of the curves have the same value of p at the point. 33. The singular solution. Suppose that f(^,y,p) = o (1) is the differential equation, which has {x,y,c) = (2) for its solution. It has been seen, in Arts. 4-6, that the system of curves which is the locus of f(x, y, p) = Q is the set of curves obtained by giving c all possible values in (2). The X, y, p, at each point on the envelope of .the system of curves which is the locus of (2), being identical with the x, y, p, of some point on one of these curves, satisfy (1). Therefore the equation of the envelope is also a solution of that differential equation. This is called the singular solution. It is distin- guished from a particular solution, in that it is not contained §33.] THE SINGULAR SOLUTION. 43 in the general solution; that is, it is not derived by giving the constant in the general solution a particular value. The singular solution may be obtained from the differential equation directly, without any knowledge of the general solu- tion. For, at the points of ultimate intersection of consecutive curves, the ^'s for the intersecting curves become equal, and thus the locus of the points where the p's have equal roots will include the envelope ; that is, the p discriminant relation of (1) contains the equation of the envelope of the system of curves represented by (2). In the next article, it will be shown that the p and c discriminant relations may sometimes repre- sent other loci besides the envelope : that is, they may contain other equations besides the singular solution. The part of these relations that satisfies the differential equation is the singular solution. E..1. , = ,* + '<^^'. which is in Clairaut's form, has for its solution y = cx-\- aVi + ti'. This, on rationalization, becomes c2 (flt - a;2) _|_ 2 czij + a2 _ j2 = 0, .^v^ ^ and hence the condition for equal roots is a;2 + 5,2 = a^. This relation satisfies the given equation, and hence is the singular solution. In this example, the general integral represents the system of lines y = cx + aVl + c% all of which touch the circle x^ + y"^ = a'. Ex. 2. Find the general and the singular solutions oi p'^ + xp — y = 0. Ex. 3. Find the general and the singular solutions of dyVx = dxVy. Ex. 4. Find the singular solution of xV — 3 xyp + 2y^ + x^ = 0. Ex. 5. Find the general and the singular solutions of ('-iy=s<-'>('-s)" 44 DIFFERENTIAL EQUATIONS. [Ch. IV. 34. Clairaut's equation. In finding tKe solution of Clairaut's form in Art. 28, there appeared the equation x+f'{p) = 0, (3) which is as important as the equation -~- = 0, that appeared with it. The foregoing shows what part equation (3) plays in solving Clairaut's equation. On differentiating y=px-j-f(p) with respect to p, (3) is obtained. The elimination of p between these two equations gives the p discriminant relation, which here represents the envelope of the system of lines y=cx+f{c) represented by the general solution. 35. ' Relations, not solutions, that may appear in the p and c dis- criminant relations. It has been pointed out that the p dis- criminant relation of f(x, ?/, p) =i= represents the locus, for , each point of which f(x, y, p) = will have equal values of p ; and that the c discriminant relation of ^ (x, y, c) = 0, the gen- eral solution of the former equation, represents the locus for each point of which ^ (x, y, c) = will have equal values of c. It is known also that each point on the envelope of the system {x, 2/, c) = is a point of ultimate intersection of a pair of consecutive curves of that system ; and, moreover, that at each point on the envelope there will be two equal values of p, one for each of the consecutive curves intersecting at the point ; and that, therefore, the singular solution, representing the envelope, must appear in both the p and the c discriminant relations. But the question then arises, may there not be other loci besides the envelope, whose points will make f{x, y, p') = give equal values of p, or make cj) (x, y,c) = Q give equal values of c ? In other words, while the p and the c discriminant relations must both contain the singular solu- tion, which represents the envelope if there be one, may they not each contain something else ? § 34-37.] TAC-LOCUS AND NODAL LOCUS. 45 36. Equation of the tac-locus. At a point satisfying the p discriminant relation there are two equal values of p; these equal p's, however, may belong to two curves of the system that are not consecutive, but which happen to touch at 'the point in question. Such a point of contact of two non-consecu- tive curves is on a locus called the tac4ocus of the system of curves. The equations representing the tac-locus, while thus appearing in the p discriminant relation, will not be contained in that of the c discriminant ; since the touching curves, being non-consecutive, will have different c's. Ex. Examine y\l+p^)=r'K Vr2 — w2 Solving for p, p = —■ Integrating and rationalizing, 2/2+(x-t-c)2= r^. The general solution, therefore, represents a system of circles having a radius equal to r and their centres on the x axis. The c discriminant relation is y'- — r'- = 0, and that of the p discriminant is i/^(!/'^ — )'^) = 0. Thus the locus of the latter is made up of the loci y = ±r and of y = counted twice. The equations y = ± r, that appear in both the p and the c discriminant relations, satisfy the differential equation, and hence form the singular solution; they represent the envelope. The equation y = 0, as is apparent on substitution, does not satisfy the differential equation. Through every point on the locus y-= 0, two circles of the system can be drawn touching each other ; that equation, there- fore, represents the tac-locus. The student is advised to make a figure, showing the set of circles, their envelope, and the tac-locus, as it will help him to understand this and the preceding articles. 37. Equation of the nodal locus. The c discriminant relation, like that of the'p discriminant, may contain an equation having a locus, the x, y, p, of whose points will not satisfy the differ- ential equation. 46 DIFFERENTIAL EQUATIONS. [Ch. IV. The general solution <^ (x, y, c) = may represent a set of curves each of which has a double point. Changing the c changes the position of the curve, but not its character. These Fig. 1. curves being supposed drawn, the double points will lie on a curve which is called the nodal locus. In the limit two con- secutive curves of the system will have their nodes in coin- cidence upon the nodal locus. The node is thus one of the ultimate points of intersection of consecutive curves ; and, therefore, the equation of this locus must appear in the c dis- criminant relation. But in Pig. 1, where A, B, ■••, are the curves and L is the nodal locus, at any point the p for the nodal locus L is different from the p's of the particular curve that passes through the point ; and hence the x, y, p, belong- ing to L at the point, will not satisfy the differential equation. Fia. 2. And, in general, the x, y, p, at points on the nodal locus will not satisfy the differential equation; for the case would be exceptional where the p at any point on the nodal locus would §38.] CUSPIDAL LOCUS. 47 coincide with a p for a curve of the general solution passing through that point; where, in other words, the nodal locus would also be an envelope, as in Fig. 2, in which A, B, •••, L, have the same signification as in Fig. 1. Ex. Kp2 — (a; — a)^ = has for its general solution y + c = 2 x' - 2 ax^ ; that is, |(j/ + c)2 = x(x-3a)2. Thep discriminant relation is x(x — a)^ = 0, and that of the c discriminant, x(x — 3 a)^ = 0. The relation x = satisfies the differential equation ; hence it is the singular solution and represents the enve- lope locus. The relation x — a = 0, which appears only in the p discriminant, does not satisfy the differential equation ; it represents the tac-lociis. And x — 3 a = 0, which is in the c discriminant, does not satisfy the original equation ; it represents the nodal locus. Figure 3 shows some of the curves of the system, the envelope, the tac, and the nodal loci. 38. Equation of the cuspidal locus. The general solution <^ {x, y, c) = may represent a set of curves each of which has a cusp. These curves being supposed drawn, the cusps will lie on a curve called the cuspidal locus. It is evident that in the limit two consecutive curves of the system will have their cusps coincident upon the cuspidal locus, the cusps thus being among the ultimate points of intersection ; and hence the cuspi- dal locus will appear in the locus of the c discriminant relation. Moreover, the p's "at the cusps of consecutive curves will evi- deutly be equal ; and therefore the cuspidal locus will appear Fig. 3. 48 DIFFERENTIAL EQUATIONS. [Ch. IV. in the locus of the p discriminant relation. Like the nodal locus, it will not, in general, be the envelope. Ex. 1. The differential equation p^ + 2xp-y = (1) has for its general solution (2 a;8 + 3 X!/ + c)2 - 4(a;2 + yy = 0. (2) The p discriminant relation is x^ + y = 0, (3) and the c discriminant relation is (a;2 + y)» = 0. Equation (1) is not satisfied by (3), and hence there is no singular solution ; a;^ ^ j^ = o is a cusp locus. Ex. 2. The equation 8 ap" = 2'7y has for its general solution ay^ = (x — c)'; I .^ .^ ^^ ^^ ^ the p discriminant relation is 2/ = 0, and the c discriminant relation is y* = 0. "' The equation y = satisfies the differential equation, and therefore is the singular solution. It is also the equation of the cusp locu^. Figure 4 illustrates this example. This is one of the very exceptional cases where the cusp locus coincides with the envelope. 39. Summary. When the loci discussed above exist, then in the p discriminant relation will appear the equations of the envelope locus, of the cuspidal locus, and of the tac-locus ; and in the c discriminant equation will appear the equations of the envelope locus, of the cuspidal locus, and of the nodal locus.* * See Edwards, Differential Calculus, Arts. 364-366 ; Johnson, Differ- ential Equations, Arts. 45-54 ; Forsyth, Differential Equations, Arts. 23-30 ; an article by Cayley, " On the theory of the singular solutions of differential equations of the first order" (Messenger of MathematicSiVol. II. [1872], pp. 6-12) ; an article by J. W. L. Glaisher, "Examples illus- trative of Cayley's theory of singular solutions" (Messenger of Mathe- matics, Vol. XII. [1882], pp. 1-14). § 39.] EXAMPLES. 49 The p discrimmant relation contains tlie equations of the envelope, cuspidal and tao loci, once, once, and twice respectively ; and the c dis- criminant relation contains the equations of the envelopfe, cuspidal and nodal loci, once, three times, and twice respectively.* EXAMPLES ON CHAPTER IV. Solve and find the singular solutions of the following equations : 1. xp'^ — 2yp + ax = 0. 3. y^ — 2px^j + p^(x'' — 1)= m^. 2. x3p2 -I- x^yp + a' = 0. i. y = xp+ VW+~^^. 8. y = xp—p^. 6. Examine Exs. 2, 4, 20, 26, Chap. III., for singular solutions. 7. Solve 4p2 = 9x, and examine for singular solution. 8. Investigate for singular solution 4x(x- l)(x-2)p-^-(3x2-6x-|-2)2 = 0. 9. Solve and examine for singular solution (8p' — 27)x = 12^^^, 10. p^{a;^ - a2) - 2pxy + y^-b^ = 0. 11. (px-y)(ix-py)=2p. * This is proved in an article by M. J. M. Hill, " On the c and p dis- criminant of ordinary iiitegrable differential equations of the first order" (Pj-oc. Lond. Math. Soc, Vol. XIX. [1888], pp. 561-589). This article supplemenis Cayley's, mentioned above. Professor Chrystal has shown that the p discriminant locus is in gen- eral a cuspidal locus for the family of integral curves. (^Nature, Vol. I.IV., 1896, p. 191.) 50 DIFFERENTIAL EQUATIONS. [Cb. V CHAPTER V. APPLICATIONS TO GEOMETRY, MECHANICS, AND PHYSICS. 40. The student will remember that, after deducing the methods of solving various kinds of algebraic equations and virorking through lists of these equations, he made practical applications of the knowledge and skill thus acquired, in the solution of problems. In the process of finding the solution of one of these problems, there were three steps : first, forming the equations that expressed the relations existing between the quantities considered in the problem ; second, solving these equations ; and third, interpreting the algebraic solution. In the case of differential equations, the same procedure will be followed. The three preceding chapters have shown methods of solving differential equations of the first order. This chapter will be concerned with practical problems, the solution of which will require the use of these methods. The problems will be chosen for the most part from geometry and mechanics; and it is presupposed that the student possesses as much knowledge of these subjects as can be acquired from elementary text-books on the differential calculus and me- chanics. As in the case of algebraic problems, there are three steps in obtaining the solution of the problems now to be considered : First, forming the differential equations that express the relations existing between the variables involved. Second, finding the solution of these equations. Third, interpreting this solution. § 40-42.] APPLICATIONS. 51 There will be only two variables involved in each of these problems, and hence but a single equation will be required. The choice of examples for this chapter is restricted, because differential equations of the first order only have so far been treated. 41. Geometrical problems. The student should review the articles in the differential calculus that deal with curves ; in particular, those articles that treat of the tangent and normal, their directions, lengths, and projections, and the articles that discuss curvature and the radius of curvature. This review will be of great service in helping him to express the data of the problem in the form of an equation, and to interpret the solution of this equation. The character of the geometrical problems and the method of their solution will in general be as follows. A curve will be described by some property be- longing to it, and from this its equation will have to be deduced. This is like what is done in analytic geometry, but here the statement of the property will take the form of a differential equation ; the solution of this differential equation will be the required equation of the curve. 42. Geometrical data. The following list of some of the principal geometrical deductions of the differential calculus is given for reference. It will be of service in forming the dif- ferential equations which express the conditions stated in the problems, or, in other words, give the properties belonging to the curves whose equations are required. Suppose that the equation of a curve, rectangular co-ordinates being chosen, is ^ ^^^^^^ ^^ j.^^^ ^^ ^ 0, and that {x, y) is any point on this curve. Then -^ is the slope of the tangent at the point (x, y), i.e. the tangent of the angle that the tangent line there makes with the a;-axis ; - — is the slope of the normal; the equation of the tangent at (a;, y), .'52 DIFFERENTIAL EQUATIONS. [Ch. V.. X, Y, being the current co-ordinates, is Y— y = ^ (^ — ^) ; and the equation of the normal is Y—y = (X — x) ; the inter- % flrg cept of the tangent on the axis of x is x — y — ; the intercept of the tangent on the axis of 2/ is y — x -^■, the length of the tangent, that is, the part of the tangent between the point and the a;-axis, is 2/^l+/'^Y; the length of the normal / ^dy\^ dx is 2/\l+(-£) ; tlie length of the subtangent is 2/^,; the \dxj ' "'^'•" "" '"" ° — """»""" ^ («2/' length of the subnormal is « -^ ; the differential of the length dx of the arc is ^1 +f^ydy, or -^1 +(^dx; the differential of the area is y dx or x dy. Again, let the equation of the curve in polar co-ordinates be /(r, 6l)=0, orr = F(e), and (r, 6) be any point on the curve. Then the tangent of the angle between the radius vector and the part of the tangent to the curve at (r, 6) drawn back towards the initial line, is r — ; if ^ is the vectorial angle, \li the angle between the radius dr vector and the tangent at (r, 6), and the angle that this tangent makes with the initial line, tj> = \f; + 6; the length of dB the polar subtangent is r' — ; the length of the polar sub- dr '^^ normal is — ; the differential of the length of the arc is d6 yjl + r'f'^ydr, or yjf^y+r''d6; if p denote the length of the perpendicular from the pole upon the tangent,* then * WiWiamson, Diffe7-e7itial Calculus, Ait. 183; Edwards, Differential Calculus for Beginners, Art. 95. §43.] GEOMETRICAL EXAMPLES. 53 that is, "2 ~ ''^ + ( :7Z ) ' '''^'isre u = ^ — = M^ + f — ) where m = — p' \dOj' r 43. Examples. Ex. 1. Determine the curve whose subtangent is n times the abscissa of the point of contact ; and find the particular curve which passes through the point (2, 3). Let (x, y) be any pomt upon the curve. The subtangent is y — . There- dy fore, the condition that must be satisfied at any point of the required curve, in other words, the given property of the curve, is expressed by the equation yf=nx. dy Integration gives n log y = log ex, whence, J/" = ex. This represents a family of curves, each of which passes through the 3" origin. For the particular curve that passes through (2, 3), c must be — , and the equation is 2y" = 3''x. When n = 1, tlie required curve is any one of the straight lines which pass through the origin ; the equation of the particular line through (2, 3) is 2y = 3x. When n = 2, the curves having the given property are the parabolas whose vertices are at the origin, and whose axes coincide with the x-axis ; the particular parabola through (2, 3) has the equation •2y^ = 9x. When ra = f , the required curve is any one of the system of semi- cubical parabolas that have their vertices at the origin and their axes coinciding with the axis of y. What curves have the given property when w = | ? When n = | ? Ex. 2. Find' the curve in which the perpendicular upon the tangent from the foot of the ordinate of the point of contact is constant and equal to a ; and determine the constant of integration in such a manner that the curve shall cut the axis of y at right angles. 64 DIFFERENTIAL EQUATIONS. [Ch. V. Let (x, y) be any point on the curve. The equation of the tangent at (x, y) is r-. = |(X-x); the length of the perpendicular from (x, 0), the foot of the ordinate, • -y upon the tangent is I I dy\^ Therefore, the given property of the curve is expressed by the equation rn ~ ^ - =a; from this, (2) — " ^ z=dx; integration gives * cosh-'- = - + c; whence (3) - = cosh ( - + c ) • It is also required that there be found the particular one of these curves that cuts the t/-axis at right angles. This means that for this curve, ^ = vfhen x = 0. Now differentiation of (3) gives dx \dy 1 . , /x , \ - / = - snih - + c ; aax a \a / therefore c = ; and hence - = cosh -, a a the equation of the catenary. Ex. 3. Determine the curve in which the subtangent is n times the subnormal. Ex. 4. Determine the curve in which tiie length of the arc measured from a fixed point A to any point P is proportional to the square root of the abscissa of P. Ex. 5. Find the curve in which the polar subnormal is proportional to the sine of the vectorial angle. Ex. 6. Find the curve in which the polar subtangent is proportional to the length of the radius vector. * See McMahon, Hyperbolic Functions (Merriman and Woodward, Higher Mathematics, Chap. IV.), Arts. 14, 15, 26, 39 ; Edwards, Integral Calculus for Beginners, Arts 28-44. § 44, 45.] TRAJECTORIES. 55 44. Problems relating to trajectories. An important group of geometrical problems is that which deals with trajectories. A trajectory of a family of curves is a line that cuts all the members of the family according to a given law ; for example, the line which cuts all the curves of the family at points equi- distant from the a;-axis, the distance being measured along the curves of the family. Another example of a trajectory is the line that cuts the curves of the family at a constant angle. When the angle is a right angle, the trajectories are called orthogonal trajectories ; when it is other than a right angle, the trajectories are said to be oblique. Only these two classes of trajectories will here be discussed. 45. Trajectories, rectangular co-ordinates. Suppose that f{x,y,a)^0 (1) Is the equation of the given system of curves, a being the arbitrary parameter ; and that a is the angle at which the tra- jectories are to cut the given curves. The elimination of a from (1) gives an equation of the form <^(«',2/,|) = 0, (2) the differential equation of the family of curves. Now through any point (x, y) there pass a curve of the given system and one of the trajectories, cutting each other at an angle «. If m is the slope of the tangent to the trajectory at this point, then -^ — tan a ax /OS m = -j \^) 1 + T^tan a dx By definition m is ~ for the trajectory ; hence the differ- ential equation of the system of trajectories is obtained by civ substituting this value of m for -^ in (2) ; this gives 56 DIFFERENTIAL EQUATIONS. [Ch. V. ^,y, dy -^ — tan a ax 1 + T^ tan a ax = 0, (4) for the differential equation of the system of trajectories ; and the solution of this is the integral equation. If a is a right angle, dx dy and hence the differential equation of the system of orthogonal this (5) trajectories is obtained by substituting for -^ in (2) ; this dy dx gives ^ Integration will give the equation in the ordi:nary form. 46. Orthogonal trajectories, polar co-ordinates. Suppose that f{r, e,c) = (1) is the polar equation of the given curve, and that (2) is the corresponding differential equation, obtained by eliminat- ing the arbitrary constant c. The tangent of the angle between the radius vector and the tangent to a curve of the given sys- dB tern at any point (r, 6) is r — . If m is the tangent of the angle between this radius vector and the tangent to the trajectory through that point, Idr m = ) rde since the tangents of the curve and its trajectory are at right angles to each other. Hence the differential equation of the §46,47.] TBAJECTOEIES. 57 required trajectory is obtained by substituting — for r— , or, what comes to the same thing, — ?•- — for — in (2) ; this gives '''■ ^^ <^('->^. -'"f)=0 (3) as the differential equation of the required system of trajecto- ries. 47. Examples. Ex. 1. Find the equation of the curve which cuts at a constant angle whose tangent is — all the circles touching a given straight line at a given point. " Take the given point for the origin, the given line for the y-a,xis, and the perpendicular to it at the point for the cc-axis. The given system of circles then consists of the circles which pass through the origin and have their centres on the x-axis ; its equation is j/2 + x-2 - 2 ax = 0, (1) a heing the variable parameter. The elimination of a gives the differen- ' tial equation of the system of circles ; namely, dy y^ — z^ dx 2xy (2) The differential equation of the system of trajectories is obtained by substituting for -^ in equation (2) the expression dy m dx n n dx and this gives on reduction (nx^ — ny^ + 2 mxy)dx + (my'' — mx^ + 2 nxy)dy = 0. (3) The integration of this homogeneous equation gives x^ + 2/^ = 2 c{my + nx), (4) c being the constant of integration ; this represents another system of circles. The trajectory is orthogonal if n = ; equation (4) then becomes x2 + j/2 = c^y, 58 DIFFERENTIAL EQUATIONS. [Ch. V. which represents the orthogonal system of circles; these circles pass through the origin and have their centres on the j/-axis. Ex. 2. Find the orthogonal trajectories of the system of curves r" sin n6 = a". Differentiation eliminates the parameter a, and gives — + r cot ne = 0, dB the differential equation of the system. The differential equation of the system of trajectories is obtained by substituting —r^— for — ; this gives dr d9 -r-— +rcotnB = 0; dr separating the variables, integrating, and simplifying, r" cos nB = c, c being an arbitrary constant ; this is the equation of the system of orthogonal curves. Ex. 3. Find the orthogonal trajectories of a series of parabolas whose equation is y^ = i ax. Ex. 4. Find the orthogonal trajectories of the series of hypocycloids 2 2 2 x^ + y^ = af . Ex. 5. Find the equation of the system of orthogonal trajectories of a 2 a series of confocal and coaxial parabolas r = 1 + cos e Ex. 6. Find the orthogonal trajectories of the series of curves. r = a + sin 5 9. Ex. 7. Given the set of lines y = ex, c being arbitrary, find all the curves that cut these lines at a constant angle 8. 48. Mechanical and physical problems. The student should read in some text-book on mechanics the articles in which the elementary principles and formulae relating to force and motion are enunciated and deduced. The truth of the following deii- nitions and formulae will be apparent to one who understands the first principles of the calculus and the principles of me- §48.] MECHANICAL AND PHYSICAL EXAMPLES. 69 chanics as set forth in elementary ■works that do not employ the calculus. If s denotes the length of the path described by a particle moving in a straight line for any period of time ; t, the time of motion, usually estimated in seconds ; and V, the velocity of the moving particle at any particular point or instant ; then will ds J — = ■«, and dt — = the acceleration of the moving particle at any point of its path. Ex. 1. A body falls from rest ; assuming that the resistance of the air is proportional to the square of the velocity, find (a) its velocity at any instant ; (6) the distance through which it has fallen. In this case the equation for the acceleration is — = ? - Kt)2, or, putting — for k, dt g dt vphence „ ^'^^^ „ = dt. g^ — vrvr Integrating, tanh-i— =nt + c; whence, — = tanh(K« + c). But c = 0, since « = w Therefore V __ds dt = S- tanh nt ; n whence, on integration, s + c 9 log cosh nt. But s = when « = 0, therefore s _ 9 ~ «2 log cosh nt. = 0; 60 DIFFERENTIAL EQUATIONS. [Ch. V. Ex. 2. , Pind the distance passed oyer in time « by a particle wliose acceleration is constant, determining the constants of integration so that at the time ( = 0, vo is the velocity and so the distance of the particle from the point from which distance is measured. Ex. 3. The velocity possessed by a body after falling vertically from rest through a distance s is found to be V'2gs. Find the height through which it has fallen in terms of the time. EXAMPLES ON CHAPTER V. 1. Determine the curve in which the length of the subnormal is pro- portional to the square of the ordinate. 2. Determine the curve in which the part of the tangent intercepted by the axes is a constant a. [Hint : Find the singular solution.] 3. Determine the curve in which the length of the subnormal is pro- portional to the square of the abscissa. 4. Find the equation of the curve for which a differential of the arc is K times the differential of the angle made by its tangent with the a:-axis, multiplied by the cosine of this angle; and determine the constant of inte- gration so that the curve touches the K-axis at the point from which the arc is measured. 5. Find the equation of the curve where the length of the perpendicu- lar from the pole upon the tangent is constant and equal to — 6. Find the equation of the system of curves that make an angle whose tangent is — with the series of parallel lines x cos a + y sin a = p, p being the variable parameter. 7. Find the orthogonal trajectories of the system of parabolas y = ax^. 8. Find the'orthogonal trajectories of the system of circles touching a given straight line at a given point. 9. Find the orthogonal trajectories of -2 + -^ — \~'^' '^^^^^ ^ '^ arbitrary. ' 10. Find the orthogonal trajectories of the series of hyperbolas xy = k". 11. Determine the orthogonal trajectories of the system of curves r" = a" cos nfff therefrom find the orthogonal trajectories of the series ■ of lemniscata r' = a' cos 2 $. 12. Find the orthogonal trajectories of Ir-] — ) cos 8 = a, a being the parameter. § 48.] EXAMPLES. 61 13. Find the orthogonal trajectories of the series of logarithmic spirals r = a^, where a varies. 14. Determine the curve ■whose tangent outs offl from the co-ordinate axes intercepts whose sum is constant. 15. The perpendiculars from the origin upon the tangents of a curve are of constant length a. Find the equation of the curve. 16. Find the equation of the curve in which the perpendicular from the origin upon the tangent is equal to the abscissa of the point of contact. 17. Find the equation of a curve such that the projection of its ordi- nate upon the normal is equal to the abscissa. •*■ 18. Find the equation of the curve in which, if any point P be taken, the perpendicular let fall from the foot of its ordinate upon its radius vector shall cut the «/-axis where the latter is cut by the tangent to the curve at P. , 19. Find the curve in which the angle between the radius vector and the tangent is n times the vectorial angle. What is the curve when m = 1 ? When n = i? 20. Determine the curve in which the normal makes equal angles with the radius vector and the initial line. 21. Find the curve the length of whose arc measured from a given point is a mean proportional between the ordinate and twice the abscissa. 22. Find the equation of the curve in which the perpendicular from the pole upon the tangent at any point is k times the radius vector of the point. 23. If -s = -=^^ ST I 1 I > find the equation of the curve, r being p^ 0^(1 — e^)\ r I the radius vector of any point of the curve, andp the perpendicular from the pole upon the tangent at that point. 24. Find the orthogonal trajectories of the cardioids r = a(l — cos 9). 25. Show that the system of confocal and coaxial parabolas y2=4a(a; + a) is self-orthogonal. — 26. Show that a system of confocal conies is self-orthogonal. 27. Find the curve such that the rectangle under the perpendiculars from two fixed points on the normals be constant. 28. Find the curve in which the product of the perpendiculars drawn from two fixed points to any tangent is constant. 62 DIFFERENTIAL EQUATIONS. [Ch. V. f 29. The product of two ordinates drawn from two fixed points on the X-axis to the tangent of a curve is constant and equal to k"^. Find the equation of the curve. 30. Determine the curve in which the area enclosed between the tan- gent and the co-ordinate axes is equal to a^. 31. Find a curve such that the area included between a tangent, the a;-axis, aud two perpendiculars upon the tangent from two fixed points on the a;-axis is constant and equal to ifi. 32. The parabola y'^. = 4 ax rolls upon a straight line. Determine the curve traced by the focus. 33. Determine the curve in which s = ax^. 34. The equation of electromotive forces for an electric circuit contain- ing resistance and self-induction is E = Ei + L^, dt where E is the electromotive force given to the circuit, S the resistance, and L the coefficient of induction. Find the current i : (a) when .E = f{t) ; (6) when E = ; (c) when E = a, constant ; (d) when E is a simple harmonic function of the time, Em sin at, where jE?„ is the maximum value of the impressed electromotive force, and u is 2 ir times the fre- quency of alternation ; (e) when E = Ei sin ut + E^ sin {hut + e). 35. The equation of electromotive forces in terras of the current i, for an electric circuit having a resistance iJ, and having in series with that resistance a condenser of capacity G, is E = Bi + \ — , which reduces on differentiation to the form di , _i_ _! dE dt BG~ M dt' E being the electromotive force. Find the current i : (a) when E =f(t) ; (6) when E=0; (c) when E = a, constant ; (d) when E = E„ sin at. 36. Given that the equation of electromotive forces in the circuit of the last example, in terms of the charge q, is ^-^dt+0' find g' : (a) when E =f(t) ; (6) when E = 0; (c) when E = a, constant ; ((?) when E= Em sin at. 37. The acceleration of a moving particle being proportional to the cube of the velocity and negative, find the distance passed over in time t, the initial velocity being vo, and the distance being measured from the position of the particle at the time ( = 0. §49.] LINEAR EQUATIONS. 63 CHAPTER VI. LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS. 49. Linear equations defined. The complementary function, the particular integral, the complete integral. Equations of an order higher than the first have now to be considered. This chapter and the next will deal with a single class of these equations ; namely, linear differential equations. In these, the dependent variable and its derivatives appear only in the first degree and are not multiplied together, their coelScients all being con- stants or functions of x. The general form of the equation is d^'-^^'d^' + ^'d^'+ +^'-^-'^' W where X and the coefficients Pj, Pj, •••, P„, are constants or d'^v functions of x. If the derivative of highest order, -^, has a coefficient other than unity, the members of the equation can be divided by this coefficient, and then the equation will be in the form (1). The linear equation of the first order has been treated in Art. 20. It will first be shown that the complete solution of (1) con- tains, as part of itself, the complete solution of If y = yi be an integral of (2), then, as will be seen on substitution in (2), y == c^yi, Cj being an arbitrary constant, is also an integral ; similarly if 2/ = .%, y = yzy---,y = Vn, be inte- grals of (2), then y = c^2,-;y= c^^ where Cj, ••■, c„ are arbi- 64 DIFFERENTIAL EQUATION S. [Ch. VI. trary constants, are all integrals. Moreover, substitution will show that 2/ = Ci2/i + C22/2+--- +c^„ (3) is an integral. If y^, y^, •■■, ?/„, are linearly independent,* (3) is the complete integral of (2), since it contains n arbitrary con- stants and (2) is of order n. If 2/ = M be a solution of (1), then y=Y+u, • (4) where T = Ci?/i + C22/2 + • • • + c,^„, is also a solution of (1) ; for the substitution of Yioi y in the first member of (1) gives zero, and that of u for y, by hypothesis, gives X. As the solution (4) contains n arbitrary constants, it is the complete solution of equation (1). The part Y is called the complementary function ; aiTd the part ?t is called the pai'ticalar integral.'f The general or complete solution is the sum of the complementary function and the particular integral. 50. The linear equation with constant coefficients and second member zero. The equation where the coefficients Pj, Pj; ■•> P„> are constants, will first be treated, t On the substitution of e""' for y, the first member of this equation becomes (m" + Pim"^^ + ••■ + P„)e""; and this will be equal to zero if m" + Pim"-'H f-P„ = 0. . (2) * See Note F for the criterion of the linear independence of the inte- grals 2/1, 1/2, —, Vn- t This use of the term particular integral is to be distinguished from that indicated in Art. 4. J The method of solving the linear differential equation with constant coefficients, shown in this article, is due to Leonhard Euler (1707-1783), one of the most distinguished mathematicians of the eighteenth century. §50,51.] LINEAR EQUATIONS. 65 This may be called the auxiliary equation. Therefore, if m have a value, say wi,, that satisfies (2), y = e"""' is an integral of (1) ; and if the n roots of (2) be jWi, m.^, ■ ■ ■ m„, the complete solution of (1) is y = cie""* + Cse"'^' -\ f- c.e"'^^ (3) Ex. 1. Solve |^ + 3^-54!/ = 0. dx' dx Here equation (2) is' m^ + 3 m — 54 = ; solving for m, m = 6, — 9. Hence the general solution of the equation is y = cie^" + 026"^''. Ex. 2. If ^ - mhi = 0, show that y = cie"" + Cae-"" = A cosh mx + B sinh ma;.* Ex.3. Solve 2^ + 5^-12x = 0. Ex.4. Solve 9^+ 18^- 16a; =0. dz^ dz 51. Case of the auxiliary equation having equal roots. When two roots of (2) Art. 60 are equal, say m, and m^, solution (3) becomes 2/ = (cj + C2) e"'"' + Cae"*" + [-c„e'""''. But, since Cj + C2 is equivalent to only a single constant, this solution will have (n — 1) arbitrary constants ; and hence is not the general solution. In order to obtain the complete solution in this case, suppose that wis = mj -f- ^ ; the terms of the solution corresponding to mj, m^, will then be y = de'"!"' + C2e<'"'+*'% which can be written y = e"^-" (ci + c^^). * See McMahnn, Hyperbolic Functions (Merriman and Woodward, Higher Mathematics, Chap. IV.), Arts. 14 (Prob. 80), 17, 39. 66 DIFFERENTIAL FQUATIONS. [Ch. VL On expanding e'" by the exponential series, this becomes _ g-"!! (^A + Bx + -'- h terms proceeding in 1 ■ ^ ascending powers of h), where A = O1 + C2, and B = cji. Now let h approach 0, and solution (3) Art. 50 takes the form y = 6'"'=^ {A + Bx) + Cse"^ H \- c„e""^ As h approaches zero, Ci and C2 can be taken in such a way that A and B will be finite. If the auxiliary equation have three roots equal to m,, by similar reasoning it can be shown that the corresponding solu- *i°^i« y = e-^''(c, + c^ + c,x^; and, if it have r equal roots, that the corresponding solution is y = e"i' (C] + Cjas H \- cX~^)* The form of the solution in the case of repeated roots of the auxiliary equation is deduced in another way in Art. 55. Ex.1. Solve f|-3f| + 42, = 0. Ex.2. Solve 1^-^-9^-11^-4, = 0. dx* dx^ dx^ dx " 52. Case of the auxiliary equation having imaginary root's. When equation (2) Art. 50 has a pair of imaginary roots, say mi = a + I'yS, mj = a — i/J (i being used to' denote V— 1), the corresponding part of the solution can be put in a real form simpler, and hence more useful, than the exponential form of Art. 50. ~ • * See George Boole, Differential Equations, Chap. IX., Art. 7. The separate integrals, e'^", xe^", ai^e"'!'', •••, are analogous to the equal roots of an algebraic equation. J §52,53.] THE SYMBOL D. 67 For, Cie'«+'^'"' + Cje'"-'^"" = e»*(c,e*^' + Cje-'^"') = e»* I Ci (cos px + ! sin /?*) + Cj (cos /8a; — z sin j3x) \ = e"'' {A cos px + B sin /3a;) = (cosh ax + sinh ax) {A cos /3a; + B sin /8a;). If a pair of imaginary roots occurs twice, the corresponding solution is y = (cj + c^) e<''+'^''' + (cj + CiX) e'-'^-'^^', which reduces to // = e«*5 (^ + AiX) cos /3a; + (5 + Bjo;) sin |8a;j. Ex.1. Solve ^+8^+25j^ = 0. The auxiliary equation is m^ + 8 m + 25 = 0, the roots of -which are m = — 4 ± 3 (' ; and tlie solution is y = e-^{ci cos 3x -i- (;2Sin 3 a:). Ex. 2. It j^- m*y = 0, show that 2/ = ci cos mx + Ci sin vix + Cg cosh roa; + C4 sinh ma;. Ex.3. Solve ^-4^ + 8^_8^+4j/ = 0. dx* da;8 da;2 dx " 53. The symbol D. By using the symbol D for the differ- ential operator — , equation (1) Art. 50 can be written * dx (D- + PiZf-' + . . . + p^) 2/ = 0, (1) or, briefly, f{I))y = 0. (2) The symbolic coefficient of y in (1) is the same function of D that the first member of equation (2) Art. 60 is of in ; and, therefore, the roots of the latter equation being m,, m^, • • ■, m„, equation (1) may be written (D-m,){D-m2)-{D-m„)y = 0. (3) Hence the integral of (1) can be found by putting its sym- bolic coefficient equal to zero, and solving for D as if it were an ordinary algebraic quantity, without any regard to its use as an operator; and then proceeding as in Art. 50 after the roots of equation (2) of that article had been found. More- over, it is thus apparent that the complete solution of (1) or (3) is made up of the solutions of * See note K, page 208, for remarks on the symbol D. 68 DIFFERENTIAL EQUATIONS. [Ch. VI. (Z> - m,) 2/ = 0, {D-m,)y = 0,.:,{D- m„) y = 0. This symbol D will be of great service. 54. Theorem concerning D. One of the theorems relating to D is, that when the coefficient of y in (i) Art. 53 is factored as if D were an ordinary algebraic quantity, then the original dif- ferential equation will be obtained when D is given its opera- tive character, no matter in what order the factors are taken. Thus, an equation of the second order g-(« + ^)| + «^. = o, when expressed in the symbolic form, is \D'-{a + /3)D + al3ly = 0; this on factoring becomes (D — a) (D — ^)y = 0. d Eeplacing Z) by — , the latter equation becomes ax dx J\dx ^J" Operating on y with B, this becomes dx and, operating on the second factor with the first. If the factors had been written in the reverse order, {D — /3)(D — a)y = 0, and expanded as above, the same result would have been obtained. It is easily shown that the theo- rem holds for an equation of the third and any higher order. It will be noted that the symbolic factors, when used as opera- tors, are taken in order from right to left. Other theorems § 54, 55.] EEPEATED ROOTS. 69 relating to D will be proven when a reference to them happens to be required.* 55. Another way of finding the solution when the auxiliary equation has repeated roots. The form of the solution when the auxiliary equation (2) Art. 50 has repeated roots can be found in another way; namely, by employing the symbol D. According to Art. 53, the solutions corresponding to the two equal roots mi of this equation are the solutions of On writing this in the form (i> — mj) |(i) — m,)2/j = and putting V for {D — m^y, the above equation becomes (Z)-m,)'y = 0, the solution of which is ■« = Cje"". Eeplacing v by its value (D-mi)^, (D — mi)y = Cie""', which is the linear equation of the first order considered in Art. 20 ; its solution is y = e"''= (Cj + CjX). Similarly the solutions corresponding to three equal roots mj are the solutions of (D-m^fy = 0, which may be written (Z)-mi)(I>-mjy2/ = 0. On putting v for {D - mify, solving for v, and replacing the value of V as before there is obtained (B — mify = Cje"'". Putting (D for (D — m^y and proceeding as before, (Z>-mi)2/ = e""'(ciX + C2), * See Forsyth, Differential Equations, Arts. 31-35, for fuller informa- tion concerning the properties of D. 70 DIFFERENTIAL EQUATIONS. [Ch. VI. the solution of which is y __ gmw (cj;2 ^ g^ _|. Cg)j where c = ^ . It is obvious that if wii is repeated r times, the correspond- iug integrals are y = e""''(Ci + Cjcc + ••• + c^a;'"'). 56. The linear equation with constant coefficients, and second member a function of x. In this article will be considered the equation the first member of which is the same as that of equation (1) Art. 50, and the second member a function of x. It was pointed out in Art. 49 that the complete integral of (1) con- sists of two parts, — a complementary function and a particular integral, the complementary function being the complete solu- tion of the equation formed by putting the first member of (1) equal to zero. The problem now is to devise a method for obtaining the particular integral. In the symbolic notation, (1) becomes f{n)y = X (2) and the particular integral is written y = X. 57. The symbolic function — — . It is necessary to define ———X, which, as yet, is a mere symbol without meaning. For this purpose it may be said : — - — X is that function of x which, when operated upon by /(-D), gives X. ■ The operator ^--— , according to this definition, is the inverse of the y(-D) operator f(D). It can be shown from this definition and Art. 54, that ■ can be broken up into factors which may be taken m any order, or into partial fractions. § 56, 57.] TBE PAUTICULAR INTEGRAL. 71 For example, the particular integral of the equation jg ± V". and this can be put in the form 1 -X. Now apply (Z> — a)(D — /3) to this, arranging the factors of the latter operator conveniently, as is allowable by Art. 54 ; this gives ^ ' • 11 and since D — a, acting upon • X, must by defini- ^ D-a D- j3 ^ tion give — -X, this becomes D — ^ • — X, which is X by the definition of . This reduction shows that the particular integral might equally well have been written X. {D-IS){D-a) Also, — ; X may be written in the form a-l3\D-a B - /Sj which is obtained by resolving the operator into partial fractions. The result of operating upon this with D'-(a + P)D + a^ is or _J_J(i)_^)X-(Z)-«)Xj; a — p and finally, X. 72 DIFFERENTIAL EQUATIONS. [Ch. VI. The statement immediately preceding this example can easily be verified for the general case by a method similar to that used in this particular instance. 58. Methods of finding the particular integral. It is thus apparent that the particular integral of equation (2) Art. 56, namely, ——X, may be obtained in the two following ways : 1 (a) The operator — — may be factored ; then the particular integral will be J^ > 11 1 -X. D — mi D — m^ D — m„ On operating with the first symbolic factor, beginning at the right, there is obtained — 1^ = e""' Ce--"'n'Xdx;* D — mi D -m^ D — m„_, J then, on operating with the second and remaining factors in succession, taking them from right to left, there is finally obtained the value of the particular integral, namely, e^ix I g("i2-™i)« j ... j e~"»' X(da;)". (&) The operator — — may be decomposed into its partial fractions ""^ ' D — thx D-m^ D -m^ and then the particular integral will have the form Of these two methods, the latter is generally to be preferred. Since the methods (a) and (6) consist altogether of operations of the kind effected by upon X, the result of the latter operation should be remembered. Now, X is the par- ticular integral of the linear equation of the first order, * This is made clear in the last paragraph of this article. § 58, 59J SHORT METHODS. ' 73 which has been discussed in Art. 20 ; its value is Xdx. "f' The term Cie'^ in the solution of this equation is the comple- mentary function. Ex. 1. Solve ^-5^ + 62/ = eK fZx^ dx This equation written in symbolic form is or (2>-3)(Z)-2)?/ = e^; I lience the complementary function isy = Cie^ + c^e^ ; and the particulai integral is " Z»-3 D-2 \D-3 D-2J and lience the genei'al solution is y = ae^ + c^e^ + I e*". Ex. 2. Solve -^^-y = 2 + bx. Ex. 3. Solve p, + 2^ + y = 2e^. _^-- Ex.4. Solve ^-f|-8^+12, = X dx^ dx^ dx 59. Short methods of finding the particular integrals in certain cases. The terms of the particular integral which correspond to terms of certain special forms that may appear in the sec- ond member of the equation, can be obtained by methods that are much shorter than the general methods shown in the last article. The special forms occurring in the second member which will be discussed here are : 74 DIFFERENTIAL EQUATIONS. [Ch. VL I (a) e"", where a is any constant ; 1 (6) x'", where m is a positive integer ; \ (c) sin ax, cos ax ; (d) e" V, where V is any function of x ; (e) xV, where V is any function of x. 60. Integral corresponding to a term of form e" in the second member. The integral corresponding to e"' in the second member of the equation /(£>)?/= X, is e""; this will be shown to be equal to — — e^. Successive differentiation gives D"e'" = a"e"'; the terms appearing in f{D) are terms of the form D", n being an integer ; therefore /(Z))e-=/(a)e-. Operating on both members with ^ -/(D)e«' = -i-/(a)e'"; and this, since — — and /(Z>) are in.verse operators and /(«) is only an algebraic multiplier, reduces to e""=/(a) -i- e""^; whence e"" = ^— e"'. The method fails if a is a root of /(Z)) = 0, for then — — e"' = 00 e"". In this case, the procedure is as follows. Since a is a root of f{D)= 0, {D - a) is a factor of /(/)) Suppose that f{D) = {p-a)<^ (D) ; then /(D) iD-a) {D) (D-a) (a) ,/, (a) " ?60, 61.] TERM OF FORM a;". 75 If a is a double root of f(D) = 0, then D — a enters twice as a factor into f{D). Suppose that /(i>) = {D — af \p {D) ; then 1 g„,^ 1 1 ^„. 1 e- a;V' The method of procedure is obvious for the case when a is a root of /(D) = 0, r times. Ex. 1. Solve yl + J/ = 3 + e-' + 5 e2». Written in symbolic form, this equation becomes (Z)3+l)2/ = 3 + e-»^ + 5e2»=. Here the roots of /(Z)) = are — 1, ^ ; hence the complementary function is The particular integral is ce-" + e2( ci cos ^^ + d sin 5^"] -D° + 1 ^^ substitution of and 2 for D, on account of the first and third terms, gives 3 + f e^. But —1 is a root of D'^ + 1, hence, fact 'ring the denominator, 1 1 .e-. = . 1 e- Z)3 + 1 D + 1 Z>2 - Z) + 1 ■ D+i 3 ' on substituting — 1 for D in tlie second factor ; tlie last expression is equal to — — ; hence the complete solution is o y = ce-" + e\ci cos"2 + ca sin2") + 3 + 5.«& + ^- Ex. 3. Find the particular integi'als of Exs. 1, 3, Art. 58, by the short method. Ex. 3. Solve ^-y=(e' + 1)^. Ex. 4. Solve ^ - 2 ^ + 2/ = 3 eh dx'' dx 61. Integral corresponding to a term of form a;™ in the second member, m being a positive integer. When x" is to be 76 DIFFERENTIAL EQUATIONS. [Ch. VI. evaluated, raise /(-D) to the (— l)th power, arranging the terms in ascending powers of D; with the several terms of the expression thus obtained, operate on a;" ; the result will be the particular integral corresponding to a;". It is obvious that terms of the expansion beyond the mth power of D need not be written, since the result of their operation on a;"* would be zero. Ex. 1. Solve (I»3 + 3 D2 + 2 D)y = x\ The roots of f{D) are 0,-2, — 1 ; and hence the complementary function is Ci + de-^" + Cje"*. The particular integral = ^(l-|iJ + fZ)2+...)a;^. = ^(»:=-3» + l) = ^(2^=-9« + 21), — z being merely i xdx. The complete solution is y = ei + de^ + cae""^ + -^ (%v?-^x\ 21). The operator on 7? could equally well have been put in the form and this gives the result already obtained. One might think that it would be necessary to add another term, IP, in this form of Che operator ; but the result for this term would be a numerical constant; and this is already included in the complementary function. Ex. 2. Solve Ex. 2, Art. 58, by this method. Ex.3. Solve 1^+8?/ = a;4 + 2x + 1. 62. Integral corresponding to a term of form sin ax or cos ax in the second member. Successive differentiation of sin ax gives D sin ax =a cos ax, D^ sin ax = — a^ sin ax, §62.] TEEM OF FORM SIN ax OB COS ax. 77 £>" sill ax = — a^ cos ax, D* sin oa; = + a* sin ax = (—a^^ sin ax, and, in general, (D^)" sin ax = (— a*)" sin ax. Therefore, if (ft (Z>-) be a rational integral function of D^, 2 _ 1)2 25 = _lsin2a;-52il^; 25 25 hence the complete solution is y = cxe- +'e-' (c^ + ca ck) - 1: sin 2 x - 5^- 25 25 The number, — 4, might have been substituted for D^ ^t any step in the work. 78 DIFFERENTIAL EQUATIONS. [Ch. VI. Ex. 2. Solve -3-| + a''y = cos ax. The complementary function is ti cos ax + c^ sin ax ; the particular integral is = cos ax = = cos ax ; and thus the method fails. 6 2,2 + a-2 -a^ + a^ In this case, change a to a + h; this gives for the value of the particular integral, cos(a + ft)x; this expression, on the application of the principle above and the expansion of the operand by Taylor's series, 1 h^x^ becomes Ccos ax — sin ax- hx — cos ax ■ 1 — ). - (a + A)2 + a^ ^ 1 • 2 • The first term is already contained in the complementary function, and hence need not be regarded here ; the particular integral will accord- ingly be written - (x sin ax -\ — ■:!^— cos ax + terms with higher powers of k) ; 2a + h 1-2. on mailing h approach zero, this reduces to —• The complete integral is y = 0\ cos ax + ci sin ax -) 2a Ex.3. Solve ^-4?/ = 2 sin iK. da? Ex. 4. Solve ^ + « = sin 3 a; - cos'' * x. dx^ " ^ 63. Integral corresponding to a term of form e'"F in the second member, V being any function of x. Since De!" V = e^D V + ae"" V=e'"{D-i-a) V; and D'e'"V= ae''%D + a)V+ e'^D (D + a)V= e-(Z> + ayV; and, in general, as is apparent from successive differentiation, jy»e'"'V= e^{D + afV; therefore, f{D)e'^V= e'^f(D + a) V. (1) Now put f(D + a)V= V, ; then F= \ Vy. Also, Fi J(,I' -r <^) will be any function of x, since F is any function of x. Sub- stitution of this value of F in (1) gives ■'^ ^ fiD + a) §0a, 64.] TERM OF FORM xV. 79 whence, operating on both members of this equation with — — — , and transposing, where Vi, as has been observed, is any function of x. Ex.1. Solve -j4 + 2/ = a;e^^ /v'^ ^ i ^ The complete solution is 2/ = Ci cos a; + C2 sin a; H — - — -xe^. D^ + 1 By the formula just obtained, ^ -xe'^ = e^ 5_ X = e^ --^ -x ; JD2 + 1 (i) + 2)2 + 1 5 + 4 jD + Z)2 and this, by the method of Art. 61, gives the integral — (5 a; — 4). The complete solution is y = ci cos a; + C2 sin a; H (5 x — 4). 25 Ex. 2. Solve ^+3^ + 2j^ = e2»sina;. dx^ dx Ex. 3. Solve -j\+2y = x^e^ + e" cos 2 x. 64. Integral corresponding to a term of the form a;F in the second member, V being any function of x. Suppose that a term of the form a; F occurs in f(D)y = X. Differentiation shows that DxV = xDV+ V, D^xV=xDW+2DV, D''xV= xD"V+ niy-W\ or, as it may be written, = xD^V+ (jf^^A V. Therefore, f{D)xV=xf{D:)V + fiD)V. (1) 80 DIFFERENTIAL EQUATIONS. [Ch. VI. The formula in the case of the inverse operator derived in the following way. ''^ ' In formula (1) put f{D) V= Vi- Then V= — ^ Fj. Since V is any function of x, Vi is any function of x. The substitu- tion of this value of V in (1) gives On operating on both members of this equation with and transposing, -AD)' f{D) . f{D) f{D) ^ ' f{D) I f{D) •'^ '\ /(D) ' The particular integral corresponding to expressions of the form K'F, where r is a positive integer, can be obtained by successive applications of this method. Constants of integra- tion should not be introduced in the process of finding par- ticular integrals. Ex. 1. Find the particular integral of Ex. 1, Art. 63, by this method. The particular integral = — - — xe^ = ( x — • • 2 Z> 1 — - — e^ ^ ^ D'+l \ D^+l ) D^ + 1 5 Z)2 + 1 5 _x^_ 1 4e^ 5 Z)2 + 1 5 = |3(5.-4). Ex. 2. Solve -~ -I- 4 M = a; sin x. Ex. 3. Solve -rrA — y = x" cos x- dx^ dx:^ " EXAMPLES ON CHAPTER VI. 1. g+4, = 0. V 2. (i35 - 13 1)8 -I- 26 2)2 -I- 82 Z» + 104)?/ = 0. §64.] EXAMPLES. ' 81 4. ^ + 4!/ = sin3a; + e^ + xV 13. f^„-2^+ y = x^e^. \^ da? ^ 1 1 » dx^ dx '' 15 ^ — a^M = K*. 6. {Tfi — a:^)y = e"" + e"''. t/' ' doti^ '• d^^-^d^^-^dx^^^-''- ^ ^®- d^*~^dx3 + dX^-''- «• -^^ + ^3 + ^. = ^^(« + &-)-^ 1^- ^.-y = e^'^sx. 22. (Z)'-2Z)3 -3i3-^ + 4/) + 4)j/=a;V. 24. -^,-y = xsinx+(l + x'^)e'^. dot!' 25. (i)2 — 4Z) + 3)2/ = e*cos2a; + cos3K. — 26. (D3_3x)2 + 4I»-2)j/ = e' + cosx. . 27. (D2_9X» + 20)!/ = 20x. '^ (Py %. ■ , 1 ■ ^^ ^ I HI — /i2k Din /*• _L a^ cm • 29. ^ + w = e^sinx + fi^sin-^ 30 Show that ND + M -^ .^j^gj.g jyaud ikf are constants, is equal (2) - a)2 + (32 to twice the real part of — ( ^ -^ X operated on by ND + M; 2i/3 \/>- a- 1/3/ that is, to ^^^iaiEje-c^cos^x • Xdx _ff^^je-a==sin^x ■ Xdx. (Johnson, Diff. Eq., Art. 106.) 82 DIFFERENTIAL EQUATIONS. [Ch. VH. CHAPTER VII. LINEAR EQUATIONS "WITH VARIABLE- COEFFICIENTS. 65. The homogeneous linear equation. First method of solution. This chapter will treat of linear equations in which the coefficients are functions of x. For the most part, however, it will discuss only a very special class of these equations, namely, the homogeneous linear equation. A homogeneous linear equation is an equation of the form ^"d+p^'' F^ + - +p«-^^£+p^ = X, (1) where p^, p^, ••■, p„, are constants, and X is a function of x. This equation can be transformed into an equation with constant coefficients by changing the independent variable x to z, the relation between x and z being z = log X, that is, x = e^ If this change be made, then dy __dy dz _1 dy dx dzdx~ X dz' da? x^ydz"" dz/ d^_ 1/d'y d^2/ da?- 0!=^^"'*^+"' d"y _ 1^ fd^y _ n-n — 1 d^'hj dx" §65.] HOMOGENEOUS LINEAR EQUATIONS. 83 On putting D for — , and clearing of fractions, these equa- ,. , dz tions become da? :D(D-l)(D-2)y, x-^^=D{D-V){D-2) {D-n + l)y; (2) and, in general, the operators only and not the operand being indicated, of— = D(D-1) daf ^ ' (i)-r + l). (3) Hence, the substitution of e' for x in (1) changes it into the form Ji)(Z>-l)...(Z>-w+l) + i>i-D(i>-l)"-(i>-w + 2)+...+i9„|2/ = Z, (4) where Z is the function of z into which X is changed. Equation (4), having constant coef&cients, can be solved by the methods of the last chapter. If its solution hey — f{z), then the solution of (1) is y^f(\ogx). Therefore equation (1) can be solved by putting x equal to e", thus changing the independent variable from x to z, which reduces the given equation to one with constant coefficients ; and then solving the newly formed equation by the methods of the preceding chapter. Ex. 1. Solve x^-. x^ + 2/ = 21oga;. On changing the independent variable by putting x equal to e', this equation becomes 84 DIFFERENTIAL EQUATIONS. [Ch. VII. On solving this equation by the methods of Arts. 51, 61, the complete integral is found to be which, in terms of x, is •y = x{ci + C2log5i;)+ 21ogx + 4. Ex. 2. Solve K^i^ + 2/ = 3x2. 66. Second method of solution : (A) To find the complementary function. The solution of can be found directly, without explicitly making the trans- formation shown in Art. 65. The first member of (1) becomes, when a;" is substituted for y, fm(m — l)(m— 2) ••• (m — n + 1) + pi'm(m — 1) ••■ (m — w + 2) + — +p„]x''. Therefore, if m(m — 1) ••• (m — n + T) + pim{m — 1) ■•• (m — n + 2) + -+Pn=0, (2) the substitution of flj™ for y makes the. first member of (1) vanish ; and then a;"* is a part of the complementary function of the solution of (1). Therefore, if the n roots of (2) are mi, m.2, ■■■,m„, the complementary function of the solution of (l)is the c's being arbitrary constants. It will be noticed that the first member of (2) is the same function of m as the coefficient of y in equation (4) Art. 66 is of D. Therefore, corresponding to an integral y = x"^ of (1), there is an integral y = e"^' of (4) Art. 65 ; and hence, as has §66,67.] SECOND METHOD OF SOLUTION. ' 85 already been seen, an integral of (1) can be obtained by substi- tuting log X for 2 in the integral of (4) Art. 65. Therefore, if (2) has a root wii repeated r times, the corresponding integral of equation (4) Art. 65 being y = e"*>'(ci + C2Z + c^z^-\ h c^-^), the integral of (1) is y = x'^lci+C2\ogx-{ \- c,Q.ogxy-^\. Similarly, if (2) have a pair of imaginary roots a ± i^, the corresponding integral of (4) Art. 66 being y = e'^(c, cos /82 + Ca sin j8«), the integral of (1) is 2/ = a^JciCOs(j81oga5)+ C2sin(|81oga;)|. Ex.1. SoUex^^,+ 3x^^„+x^+y = 0. dz^ dx^ dx Substitution of k™ for y gives {m" + l)x'» = ; the roots of this equation are — 1, -= — —; and hence the solution Is V = -1 + K^l C2 cos^^logx^ + C3 sin ^^loga;H . Ex. 2. Find the complementary functions of Exs. 1, 2, Art. 65, by this method. Ex. 3. Solve K*^ + 6x3^1+ 9x2|^+ 3a;^ + 2/ = 0. dx* dx^ da? dx 67. Second method of solution : (B) To find the particular inte- gral. In this article and the two following, the particular integral of will be found. 86 ■ DIFFERENTIAL EQUATIONS. [Ch. VII. Since the symbol D in (3) Art. 65 stands for — , that is, for X — , this equation may be written dx . d' d f d ^\ f d ,-,' a;' — = X — X 1 ■■•( X r + 1 dx^ dx\ dx J \ dx therefore (1), when 6 is substituted for x — therein,! takes the form ^^ The coefficient of y here i^' the same function of 6 as the first member of (2) Art. 66 is of m. Let this equation be written in the symbolic form my = x. (3) The method deduced in Art. 66 for finding the comple- mentary function of (3) may on making use of the symbol 6 be thus indicated : If the n roots of f(6) = are 6i, O^, ■ • ■, d„, the complementary function of (3) is CyX^' + c^^^ + ■•• + C„!B*», the c's being arbitrary constants. The particular integral of (3), after the manner used in the case of f(D)y = X, in the last chapter, may be expressed in the form -~—X. A method for evaluating ^— X must now be devised. 68. The symbolic functions f{B) and — -. As to the direct symbol f(ff), it is to be observed that its factors are commuta- tive. For example, * See Forsyth, Differential Equations, Arts. 36, 37. t Thus e stands for the — of Art. 65, which was there symbolized by dz . , D ; but as D had already been used to indicate — , this new symbol is required. ^ §68,69.] THE SYMBOL C FUNCTION f (6). 87 and (6 - a){6 - P){6 - y)y = a^^+ (3 - a- ^ - y)^^g + («/3 + ^y + y«- « - ^ - y + l)x^- ajSyy = 0; and this shows, by the symmetry of the constant coefficients, that the order of the operative factors is indifferent. The student can complete the proof of this theorem concerning f(6) for himself. If ---- X be defined as that function of x which when operated upon by f(6) will give X, then it can be shown, by the method followed in Art. 57 in the case of the symbolic func- tion , that can be decomposed into factors which are f{0)\ fie) ^ commutative ; and also that it can be broken up into partial fractions. 69. Methods of finding the particular integral. It is thus apjjarent that the particular integral of (3) Art. 67 can be found in the following ways : (a) The operator — — may be expressed in factorial form, and X will then become m 1 1 1 ^. — . — . .. _ — _^ ^ 6 — tti 6 — a2 a — a.„ here the operations indicated by the factors are to be taken in succession, beginning with the first on the right ; the final result will be the particular integral. (6) The operator — — may be broken up into partial frac- tions, and consequently — — X be thus expressed, 6 — tti — Ui 9 — a, 88 DIFFERENTIAL EQUATIONS. [Ch. VII. the sum of the results of the operations indicated will be the particular integral. Since the methods (a) and (6) are made up of operations of the kind effected by upon X, the result of the latter 6 — a operation should be impressed upon the memory. Now, — - — X is the particular integral of the linear equar 6 — a tion of the first order x^-ay = X. dx The particular integral of this equation, by Art. 20, is found to be x^ ( x''^-^Xdx; therefore — =—X=x'^ Cx-'^-^Xdx. 6 — a J Hence the method (a) will give xh j a!'^-«i-i I ... j a;""-"" '~^X(da;)" as the value of the particular integral of (3) Art. 67 ; and the method (6) will give N^x^ Cx-'i-^Xdx+ ^^2 Cx-'i-^Xdx + ■■• +N„xf^ i x-'»'-^Xdx as its value. When a term is one of the partial fractions of (e - a)"* ' (&), the operator — - — must be applied to X, m times in suc- cession ; and this will give the result r" j arM a;-M ■ ■ ■ j a;-«-'X(da;)" Ex.1. Solve a;2^_2a;^-4« = a;*. d-j? dx Here /(9) is 6(9 - 1) - 2 9 - 4, which reduces to (9 - 4)(e + 1). § 70.] TERM OF FORM x". Hence the complementary function is cix*+~, and the particular 1 * integral is cc'. (e-4)(»+l) On using partial fractions, the latter will be written - I X* x» ) ; this reduces to b\e-4: e+1 I \x*\x-* ^+Hx — I a;-M x' ' ^+^dx, which on integration is X* log a: x^ 5 25' The complete integral is, therefore, j, = c,x* + -+-^; the term ^ is included in the term Cix* of the complementary functio* Ex.2. Solve x2^+5x^+4u = x*. dx^ dx Ex.3. Solve x2^+2x^-20!/=(x + l)2. dx^ dx n ^ ^ ^ 70. Integral corresponding to a term of form a;" in the second member. In the case of the homogeneous linear equation, as in that of the equation discussed in the last chapter, methods shorter than the general one can be deduced for finding the particular integrals which correspond to terms of special form in the second member. For instance, Proof: I x— Jx" = mx™, ( x-=- ) x" = m%"*, and for any positive integer r, ( x-y- ) x" = m'x"'. Now/(9) is a rational integral function of 6, and therefore /(S)x™ =/(m)x»>. Applying to both members of this equation, and transposing, fim) being merely an algebraic multiplier, fW fim) 90 DIFFERENTIAL EQUATIONS. [Ch. VII. If m is a root of /(fl) = 0, then f(m) = ; and hence the method fails. In this case /(d) can be factored into (9 — m)0(d); the particular integral then becomes x", which reduces to • «"■: this is - m 2 - 3 xZ) + 4)2/ = x". 10. (x*D* + 6 x3Z)3 + 9 x2Z>2 + .3 xZ» + 1)2/ = (1 + log x)^. 12. (x-^Z>2 + 3xZ» + l)|/= ^ (1 - xy 13. x''/)2 -(2m- \)xD + (to2 + re^)?/ = Ji^^™ logx. »■ ''S-''-i^-" '°"-"°r"" -- 92 DIFFERENTIAL EQUATIONS. [Ch. VIII. CHAPTER VIII. EXACT DIFFERENTIAL EQUATIONS, AND EQUA- TIONS OF PARTICULAR FORMS. INTEGRATION IN SERIES. 72. In this chapter linear differential equations that are exact will be first discussed, and then equations of certain par- ticular forms will be considered. Some of the latter come under some one of the types already treated ; but in obtaining their solutions, special modifications of the general methods can be employed. It often happens that an equation at the same time belongs to several forms ; instances of this will be found among the examples in Arts. 76, 77, 78, 79, 80. 73. Exact differential equations defined. A differential equation, ■^ \d!lf' ' dx' y) ~ ' is said to be exact when it can be derived by differentiation merely, and without any further process, from an equation of the next lower order /(£*■■•■ |.»)=/^^+' Exact differential equations of the first order have been treated in Art. 11. 74. Criterion of an exact differential equation. The condition will now be found which the coefficients of a differential equation, „ „_,^ ^«5/» + ^^d.^ + - + ^"^ = 0' (^) § 72-74.] EXACT DIFFERENTIAL EQUATIONS. 93 must satisfy in order that it be exact. The coeflB.cients Pqj Pi,---, P,„ are functions of x; in what follows, their successive derivatives will be indicated by P', P",---,P'">. The first term of (1) is evidently derivable by direct differen- ce""^?/ tiation from P^ ^ , which is therefore the first term of the integral of (1) ; on differentiating this and subtracting the result from the first member of (1), there remains The first term of (2) is evidently derivable by differentiation from (P, — Pa') -, which is therefore the second term of the integral of (1). On subtracting the derivative of this term from (2) there remains The first term of this expression is derivable from (P,-A' + P„")|^, which will therefore be the third term, of the integral of (1). By continuing this process, the expression 1 P.-1 - Pn-.' + - + (- l)»-'Po'»-« I i + ^n^ (3) will be reached, the first term of which is evidently derivable from {P„-l - PnJ + - +(- l)"-^Po"'-"|y- (4) Both terms of (3) will be derived from the expression (4), if the derivative of the coefficient of (y) in (4) be equal to P„, that is, if P„-P„-i'+-+(-l)"n<"> = 0. (5) But if both terms of (3) are derivable from the expression (4), an integral of (1) has now been obtained in the form 94 DIFFERENTIAL EQUATIONS. [Ch. VIII, + - + l^n-i - P.-.' +•••(- l)»Po'"-"|2/ = Ci. (6) Therefore (1) has an integral (6), that is, (1) is exact if its coefficients satisfy condition (5). 75. The integration of an exact equation; first integrals. If the second member of (1) Art. 74 be f(x), and condition (6) be satisfied, the second member of (6) will be if(x)dx + c. If equation (6) Art. 74 is also exact- its integral can be found in the same way; and the process can be repeated until an equation of the second or higher order that is not exact is reached; in some cases, this process can be carried on until a value of -ii, or of y is obtained. Equation (6) is called a. first integral of (1) Art. 74. It is easily proven by means of Arts. 3, 4, and Note C, p. 194, that any equation of the nth order has exactly n independent first integrals.* [See note, page 108.] Sometimes equations that are not linear can be solved by the 'trial method' employed in the case of (1) Art. 74, for instance, Exs. 6, 7, below. Ex.1. Solve a;^ + (a:2- 3)^+4x^^+2 !/ = 0. This is neither a homogeneous linear equation, nor one having constant coefficients. In it, P^ = a;. Pi = a;'' — 3, P2 = 4 x, P3 = 2 ; and the con- dition that it be exact is satisfied by these values. Integration gives The condition under which this equation is exact is also satisfied ; inte- grating again, a;J + (2;2-5)?/ = cia: + C2. * See Forsyth, Differential Equations, Arts. 7, 8. § 75.] EXAMPLES 95 This is not exact, but it is a linear equation of tlie first order, and hence solvable by the method of Art. 20. The solution is XS z2 x3 Ex. 2. Solve «*^ + 3 x"^ + (3 - 6 x)x^y = x* + 2x-5. The coefficients of this equation do not satisfy condition (5) Art. 74 ; and hence it is not exact. However, an integrating factor k" can be deduced, which will render it an exact differential equation. Multiplication by x" gives a^+^l^ + 3 a;3+>»|^ + (3 - 6 x)x''+'»y =(x* + 2x- 5)x'". Application of condition (5) Art. 74 to the first member, shows that for that condition to hold, m must satisfy the equation (m + 2) (m + 7)a;»'+3 _ 3 (m + 2)x'»+2 = ; that is, m must be equal to — 2. On using the factor — , the original equation becomes x^ which is exact. Integration gives the first integral x^^+5xil-x}y='^ + 2\ogx+l, y linear equation of the first order. Ex.3. Solvea:^+2a;^+22/ = 0. dx^ dx Ex. 4. Solve ^, + 2e'^^ + 2e'y = x\ . dx^ dx " Ex.5. Solve VS^ + 2*^ + 32/ = a;. ^ dx^ dx dxdx^ "dx' Ex. 7. Solve x22,g + (a;g - 2,)'- 3 2,^ = 0. 96 DIFFERENTIAL EQUATIONS. [Ch. VIII 76. Equations of the form -T~=f(x). This is an exact differential equation. Integration gives — — ^ = I f(x) dx + Ci; a second integration gives ^2 = fffi^Xdxf + c,x + c, ; by proceding in this way, the complete integral y = ff- ffi'^)id^y + «!»'""' + «2»" '+•••+ «»-!« + a„ is obtained. • Ex. 1. Solve -y^ = xe'. Integrating, -=-| = xe' — e' + Ci, • dv -2. = a;e» — 2 e* + cix + ca, dx y = se* — 3 e* + cx^ + CaX + Cj. This equation could also have been solved by the method of Chap. "VI. Ex. 2. Solve -^ — x". dx" Ex. 3. Solve x2^+l=0. dx* Ex. 4. Solve - -| = a;2 sin s. 77. Equations of the form ^—/(v)- -^^ equation of the form which in general is not linear and is not an exact differential equation, can be solved in the following way : cZw Multiplication of both sides by 2 -^ gives § 76-78.J EQUATIONS NOT CONTAINING y. 97 ir^tegrating, {^ = 2j/(y) cly + c,. From this, -^ = dx, whence j = x + c,. 'ffiy)dy + Ci d^'y Ex. 1. Solve ■T^+a''-y = 0- Multiplying by 2^, 2 ^ ^, = -2 a^2/^; integrating, ( -^ j = - a^j/^ ^ ci = a''{c^ -'ip') on putting a'c'^ for C\ ; dv separating the variables, = a dx, Vd' — y% V and integrating, sin ' - = ax + Cj, hence, !/ = c sin (ax + ci) ■ The given differential equation is linear, and y can be obtained directly by the method of Art. 52. The roots of tlie auxiliary equation being ± ia, the solution is ^ ^ ^^ ^.^ ^^ ^ ^^ ^^^ ^^ . that Is, 2/ = cisin (ax + Ca). This equation is an important one in physical applications. Ex.2. Solve ^ = ^- dx^ Vo!/ Ex.3. Solve ^ + ^ = 0. dx^ y^ Ex. 4. Solve -T^,-a^y = 0. dx^ 78. Equations that do not contain y directly. The typical form of these equations is 98 DlFFEIiMNTIAL EQUATIONS. [Ch. VIII. Equations of this kind of the first order were considered in Art. 26 ; and those of A.rt. 76 also belong to this class. If p be substituted for % then ^ = ^,.., f^ = f^; and (1) takes the form an equation of the (n — l)th order between x and p ; and this may possibly be solved for p. Suppose that the solution is theii y = \ F(3:) dx + c. If the derivative of lowest order appearing in the equation d'^v d^v be -^, put —^=p, find p, and therefrom find y by Art. 76. E..1. solve ..g-4.g.6| = 4. Putting p for -^, this becomes integrating, p = CiX^ + dx^ + |, whence y = ax^ + 6a;* + | a; + c. Ex. 2. Solve (sr-- Ex. 4. Solve 2 a: -t4 t4 79. Equations that do not contain x directly. The typical form of these equations is § 79, 80.] EQUATIONS NOT CONTAINING X. 99 Equations of this kind of the first order were considered in Art. 26; and those of Art. 77 also belong to this class. If p be substituted for ~, then dx and (1) will take the form an equation of the (n — l)th order between y and p which may possibly be solved for p. Suppose that the solution is p = f{y) ; then the solution of (1) is Ex. 1. Solve ^ m --•->-^s-(ir-- Ex.3, solve ,g-(|y= , nog,. Ex 4 Solve ^ + 2- + ii'^-] = by the method of this article, dx^ dx \dx/ and by that of Art. 78. 80. Equations in which y appears in only two derivatives whose orders differ by two. The typical form of these equa- tions is fd'^y dr^ ^\ (1) •^VdF"' d^" 7 ^ 100 BIFFEBENTIAL EQUATIONS. [Ch. VIII. If q be substituted for „j^ , then -r^ = ^ \ and (1) becomes from which q, that is, -^-s^, may be found. Suppose that the solution is dx»- : <^ (X), then y can be found by the method of successive integration shown in Art. 76. Ex. 1. Solve ^ + a2^ = both by this method and that of Chap. VI. Ex. 2. Solve -j^^ ~ m' -t\= e'" both by this method and that of Chap. VI. Ex.3. Solve a:2f^ + a2^ = 0. 81. Equations in which y appears in only two derivatives whose orders differ by unity. The typical form of these equa- tions is M dr;^ \_Q .J. d^~~^v d^v do If q be substituted for -:= — -. , then -^ = -^ ; and (1) becomes dx"-" dx" dx' ^ ■' an equation of the iirst order between q and x; its solution will give the value of q in terms of x. Suppose that the solu- tion is § 81, 82.] INTEGRATION IN SERIES. 101 then, from this relation, by successive integration, the value of y can be deduced. Ex. 1. Solve a''f^^ = x. On putting q for ^, this becomes dx ' integrating, aq = a^ = Vx" + c^ : ax integrating again, ay = \ [iVa^ + c^ + c^ log (a; + Vx^ + c^) + C2]. — -"S=['+(2r]* Ex. 3. Solve a;^ + f^=0. Ex.4. Solve^.f| = 2. dx^ dx^ 82. Integration of linear equations in series. When an equa- tion belongs to a form which cannot be solved by any of the methods hitherto discussed, recourse may be had to finding a convergent series arranged according to powers of the inde- pendent variable, which will approximately express the value of the dependent variable. For the purposes of this article it is assumed that such a series can be obtained.* Suppose that the linear equation can have a solution of the form y = A^ + ^iJB"" + A^ -\ h A.oT' -\ , (2) where the second member is a finite sum or a convergent series for some value or values of x. Concerning this series three things must be known : namely, the initial term, the relation * See Note B. Also Forsyth, Differential Equations, Arts. 83, 84. 102 DIFFERENTIAL EQUATIONS. [Ch. VIII. between the exponents of x, and the relation between the coefEcients. The following examples show how these things can be determined:* Ex. 1. Solve (x - a;2) f^ + 4 ^ + 2 J, = 0. (1) The substitution of x™ for y in tlie first member gives m(TO + 3)x'»-i-(m -2)(m+ l)x'». (2) This shows that, in the case of a series in ascending powers of x which is a solution of (1), tlie difference between the successive exponents of X is unity. (The difference between successive exponents may be denoted by s. ) Hence, the required series has the form ^ox" + ^ix^+i + ••• + ^r-ix^H-'-' + A^'^+'' +■■■. (3) The substitution of (8) for y in the first member of (1) gives Aomdm + 3)x"'-'+ [Ai(m + l)(m + i)-Ao{m - 2) (m + 1)] x'"+ - ) ,^, + [Arim + r)(m + r+ 3) - A-i(m + ?■ - 3)(m + ?-)]x'»+'-i+ ••.-. < When (3) is a solution of (I), the expression (4) is identically equal to zero, and the coefficient of each power of x therein is equal to zero. Therefore, on equating the coefficients of x"-' and x^+'-i to zero, it follows that m = 0, OT = - 3, A- = "^ + '' - ^ Ar-i. (6) m + )• + 3 The results (5) give the initial exponents of x and the relation between successive coefficients in the series which satisfy (1). Hence, the required series are completely determined. Hence the corresponding series is Aoil-ix + ^x^). For TO = -3, 4/=?l^^,_i. r Then Ai=-6 Ao, Ai --= 10 Ao, ^3 = - 10 ^0, ^4 = 5 -40, .46 = -^0, Ae=Aj=-- = 0. Hence the corresponding series is .4ox-3(l-5x + 10x2-10x8 + 5x*-x5). * Note L contains a general discussion to be read after Exs. 1, 2. For m = 0,^ = ^-4,-1. r + o Then At, = ■—-;; Aq = — \ Aa, ^^-2 + 3^^ ^^°' ^-3;^^="' ^. = ^-J^a = 0, ^5 = ^8=... =0. §82.] INTEGRATION IN SERIES. 103 In each of these series, Ao is an arbitrary constant, hence a solution of (1) is y = A{1 ~ ix + -^x^) + Bx-^{1 - bx + lOx'^ - lOx^ + 5x> - x^). (6) This is a general solution, since it contains two arbitrary constants. The expression (2) also shows that, in the case of a series in descend- ing powers of x which is a solution of (1), the difference between the successive exponents is — 1. Such a series has the form Aax'" + Aix"-^ + — h Ar-ix'»''-+'' + A,x'"-'- + ■■■. Substitution of this in the first member of (1) gives -(to- 2)(m+ l)AoX'"+ [Aom.(m. + 3) -Aim(m - 3)'}x''-' -\ + [^'-'(m -r + l)(m — r + 4) — Ar(m — r — 2){m — r + l)]a;'»-'-+ .... On equating the coeiBoients of x™, x"-'' to zero, it follows that m = 2, TO=-1, .4, = " ~ '' + ^ A'--', m — r — 2 On deducing the series corresponding to these values of m in the manner shown above, it will be found that a general solution of (1) is y=Ax\\-bx-'^ + \Ox-^-Wx-^+bx-^-x-^) + Bx-\l-\x-'^ + ^x-^). The first of these series is the same as the second in (6) above. The procedure when the second member of (1) is not zero vfill be made clear in the first example below. Ex. 2. Integrate (1) x*^ + a;^ + j/ = x-i. dx'^ dx First find the complementary function. The substitution of x" for y in the first member gives (2) m(m - l)x»'+2 + (to + l)*™ ; whence s= — 2, and to = or 1. r=xei The substitution of 2 AyX'^-*' for y gives r=0 '2°[(to - 2 ?•) (m - 2 J- - l)^,x"'-2r+2 + (m _ 2 r + l)^l,.x"-2'-] = 0. The coefficient of x'^-^'+^ must vanish ; therefore (m-2»-)(m-2r-l)^+ (m- 2J--I- 3)^,-1 = 0, 104 DIFFERENTIAL EQUATIONS. [Ch. VIII. hence (3) A. = ^ ^L:=2r^3^^_^_ which is the relation between the coefficients. ^°^'" = "' ^ = 27^)^-'' hence Ai= -Ao, it • o ^3 = g^^z = - Yr^«' 6t°- ; the corresponding series is 1 - J-a;-2-lx-i -l-l§x-6 -Lllilai-s _ .... 31 6! 7! 91 rorro = l, Ar= ^;-^ A-x; 2r(2r — 1) 2 — 4 hence .4i = Ao = — .4o, 2(2-1) ' Ai= ^~^ Ax = 0, 4(3-1) ^ and Ai, At, •■•are each equal to zero; the series in this case is finite, being X Hence the complementary function is V 31 5! 7 1 I \ x) In order to find the particular integral, substitute Aitx'^ for y ; then must m{m — l)Aox'"+' = x~^ ; comparison of the exponents shows that m = — 3 ; and hence Ao = tV- For m = — 3 ; the relation (3) between the coefficients becomes '"(2r + 3)(2)- + 4) '"'' hence Ai=~^Ao, ^2 = — ^^ — ^0, ^3 = llAll Ao, • ••, 5.6 5.7-6.8 5.7.9.6.8.10 § 83.] EQUATIONS OF LEGEND RE, BESSEL, ETC. 105 and the particular integral is 12V'+5.6'^ +5.7.6.8* + j' that is, 2x-^(l+—x-^ + ^-^x-* + —\ V 6! 8! ) Ex. 3. Show by the method of integration in series, that the general solution of y-| + 2/ = is ^ cos a; + B sin a;. Ex.4. (2.^ + l)g + x|+2, = 0. Ex.6. (I_cc2)g_2a;g + n(« + l)2, = 0. 83. Equations of Legendre, Bessel, Riccati; and the hjrpergeometric series.* A fuller discussion of integration in series than is here attempted is beyond the limits of an introductory course in differential equations. The purpose of Art. 82 has merely been to give the student a little idea of a method -which is of wide application ; and which is used in solving four very important equations that often occur in investigations in applied mathematics, — the equations of Riccati, Bessel, Legendre, and the hyper- geometric series. Johnson's Differential Equations, Arts. 171-180, discusses the methods to be followed when two roots of (6) Art. 82, become equal, the corre- sponding series then being identical ; and when two of the roots differ by a multiple of s, one series then being included in the other ; and when a coefficient A, is infinite. The equations referred to above, and references to be consulted con- cerning them, are as follows : t Legendre''s equation is * In connection with this article, the student is advised to read W. E. Byerly, Fourier's Series and Spherical Harmonics, Arts. 14-18. t Adrien Marie Legendre (1752-1833) was the author of Elements of Geometry, published in 1794, the modern rival of Euclid. He is noted for his researches in Elliptic Functions and Theory of Numbers. He was the creator, with Laplace, of Spherical Harmonics. 106 DIFFERENTIAL EQUATIONS. [Ch. VIII. d_ clx {i^~«^')%} + n{n + l)y = 0, where n is a constant, generally a positive integer. See Ex. 5, Art. 82. (Forsyth, Diff. Eq., Arts. 89-99; Johnson, Diff. Eq., Arts. 222-226; Byerly, Fourier's Series and Spherical Harmonics, Arts. 9, 10, 13 (c), Id, 18 (c), and Chap. V., pp. 144-194 ; Byerly, Harmonic Functions (Merriman and Woodward, Higher Mathematics, Chap. V.), Arts. 4, li>-17.) * BesseVs equation is ^^S + ^l + (^^-"^)^ = «' in which n is usually an integer. (Forsyth, Diff. Eq., Arts. 100-107 ; Johnson, Diff. Eq., Arts. 215-221 ; Byerly, Fourier's Series, etc., Arts. 11, 17, 18 (d), and Chap. VII. , pp. 219-233 ; Byerly, Harmonic Functions, Arts. 5, 19-23 of Chap. V. in Higher Mathematics; Gray and Mathews, Bessel Functions and their Applications to Physics; Todhunter, Laplace's, Lame's, and BesseVs Functions.) \ Biccati's equation is -^ + bu^ = cx^, dx to which form is reducible the equation k -^ — ai/ + 6;/' = ck". The latter equation is integrable in finite terms when n = 2a, or when ^-^ — - 2 11 is a positive integer. Riccati's equation can be reduced to a linear form, but of the second order, — ± a'^x'"u = 0. d-ji^ (Forsyth, Diff. Eq., Arts. 108-111 ; Johnson, Diff. Eq., Arts. 204-214; Glaisher, Memoir in Phil. Trans., 1881, pp. 759-828.) * Frederick Wilhelm Bessel (1784-1846) may be regarded as the founder of modern practical astronomy. In 1824, in connection with a problem in orbital motion, he introduced the functions called by his name which appear in the integrals of this equation. t Jacopo Francesco, Count Riccati (1676-1754) is best known in con- nection with this equation, which was published in 1724. He integrated it for some special cases. §83.] EXAMPLES. ■ 107 * The differential equation of the hypergeometric series is (Py . 7-(tt + j3 + l)x dy a/3 da;2 ''" a;(l - x) dx x{\ -xy~ This equation has the hypergeometric series o^ a(a+lM^ + l)^, .. 1 -7 1-2. 7-7 + 1 usually denoted by ^''(a, j3, 7, x), for one of its particular integrals ; and has a set of 24 particular integrals, each of which contains a hypergeo- metric series. (Forsyth, Diff. JSq., Arts. 113-134 ; Johnson, Diff. Eq., Arts. 181-203.) EXAMPLES ON CHAPTER VIII. 1. Show that the following equation is exact and find a first integral. [y'-^^'%)%-^^y^^Af:)-4x-y=- (Py a?- dy _ x^ 2. dx2 x{(fi - !c2) dx a^a"^ - x") 4. Find a first integral of x^g + 4 x^g + x{x^ + 2) ^ + 3 x^j/ = 2 x. - (iy-'S-{(g)^°-(gy}' * This is also called the Gaussian equation, and the series, the Gaussian series, after Karl Friedrich Gauss (1777-1855), who is regarded as one of the greatest mathematicians of the nineteenth century. He is especially noted for his invention of a new method for calculating orbits, and for his researches in the Theory of Numbers. It was Euler (see footnote, p. 64) who discovered the series and set forth its differential equation ; but Gauss made important investigations concerning the series, and showed that the ordinary algebraic, trigonometrical, and exponential series can be represented by it. (For illustrations of the last remark, see Johnson, Differential Equations, Ex. 1, p. 220.) 108 DIFFEBENTIAL EQUATIONS. [Ch. VIIl. 9. n-x^)^-x^ = 2 11 x^-x'^-^ = dx:' dx, ' ' dx? dx^ dx 12. ^ = 1 ax^ X 13. ,(l-log,)g + (l + log,)(|)^0. 14. f^ = sin2.. 15. f^ + ^^=e^. 16. -f4 + cosx^— 2sin x-^ — Mcosa = sin2a;. da;' da;2 da; " n.sin.xg = 2. lS.«g = |. ^ 19.,3g.„. 20. Find three independent first integrals of -=-| =/(»). Note for Art. 75. A first integral of a differential equation is an equation deduced from it of an order lower by unity than that of the original equation and containing an arbitrary constant. First integrals of -^ + y = are (r^l + y^ = A, -^C08x + ysinx = B, —^smx + ycosx=C, \dxj dx dx ^ = ycot(x + a). These are not all independent, for the four conetants A, B, C, a, are connected by the equations B = Acosa, C= A sin o. The elimination of -^ from the second and third of these integrals gives the solution % = js sin x + C cos ^ ; and the elimination from the first and fourth gives another form of the solution, namely, y _ ^^ si„ (a; + a). In general, if independent first integrals equal in number to the order of the equation have been obtained, all the differential coefBcients can be eliminated from them so as to leave the primitive. § 84, 85.] EQUATIONS OF THE SECOND ORDER. 109 CHAPTER IX. EQUATIONS OF THE SECOND ORDER. 84. There are other methods of solution, different from those shown in the last three chapters, which are applicable to some equations of the second order ; Arts. 85-89 will be taken up with an exposition of three of these methods. If a differential equation is not in a form to which any of the methods already described apply, it may be possible to put it in such a form. The very important transformations of an equation that can be effected by changing the dependent or the independent variable will be discussed in Arts. 90-92. Art. 93 will contain a synopsis of all the methods considered up to that point which may be employed in solving equations of the second order. 85. The complete solution in terms of a known integral. A theorem of great importance relating to the linear differential equation of the second order, is the following : If an integral included in the complementary function of such an equation be known, the complete solution can be expressed in terms of the known integral. Suppose that ?/ = 2/1 is a known integral in the complementary function of g+p|+«, = X; (1) then the complete solution of (1) can be determined in terms of 2/1. Let y = yiv 110 DIFFERENTIAL EQUATIONS. [Ch. IX. be another solution of (1) ; v will now be determined. On substituting y^u for y in (1), it will become dar \ yi ax Jdx y^ since, by hypothesis, On putting p for -^, (2) becomes dx d^+\^ + y,l^)P-J,' (^) and this equation, being linear and of the first order, can be solved for p. On using the method of Art. 20, the solution is found to be dx y^ yi' J whence, integrating, v = c, + c, f^ dx + r^ fy,eS''^X{dxy. J yi J yx J Therefore another solution of (1) is y = y,v = c^, + c,y, r^ dx + 2/1 f^ fyiel'-^XCda;)'. (4) •y ju, J y, J This includes the given solution y = yi-, and, since it con- tains two arbitrary constants, it is the complete solution. From the form of the solution (4), it is evident that the second part of the complementary function is 2/1 ) — j- dx, /g-jPdx p — T- I «ieJ''*'X(da;) 5''''' .1- , „ §86,87.] SOLUTION FOUND BY INSPECTION. Ill 86. Relation between the integrals. It is easily shown, that a y = y^, y = y^ be two independent integrals of then 2/1^-2/2^=06-^^"^ ^' dx ^^ dx (See Forsyth's Diff. Eq., Art. 65; Johnson's Diff. Eq., Art. 147.) It may also be remarked in passing, that the deduction of (3) Art. 85 from (1), when an integral of the latter is known, is an example of the theorem : that, if one or several indepen- dent integrals of a linear equation be known, the order of the equation can be lowered by a number equal to the number of the known integrals. (See Forsyth's Diff. Eq., Arts. 41, 76, 77.) 87. To find the solution by inspection. Since the complete integral of (1) Art. 85 can be found if one integral in its complementary function be known, it is generally worth while to try whether an integral in the latter can be determined by inspection. Ex.1. Solvea;^ + (l-a;)^-3/ = e^ Here, the sum of the coefBcients being zero, e' is obviously a solution of Substitution of ve' for y in the original equation gives this, on substituting p for — , becomes _-» .| + (l+.)p = l, a linear equation of the first order. Its solution is d,x XX 112 DIFFERENTIAL EQUATIONS. [Ch. IX. hence v = log a; + cj I x-'^e-'dx 4- c^ ; and therefore the complete solution is y = e" log X + Cig' \ x-^e-'dx + c^e". Equation (4) Art. 85 might have been used as a substitution formula, but it is better to work out each example by the same general method by which (4) was itself derived. Ex. 2. Solve ^^ - x^^ + xy = x. dx' dx [Here, y — xis obviously a solution when the second member is zero. A solution can often be found by an inspection of the terms of lower order in the equation.] Ex.3. Solve (3-a;)g-(9-4a:)g+(6-3a;)2/ = 0. Ex. 4. Solve x^-^ + x^ — y = 0, given that x + - is one integral. 88. The solution found by means of operational factors. Sup- pose that the linear equation of the second order is expressed in the form f{D)y = X. Sometimes f(D) can be resolved into a product of two fac- tors Fi{D) and Fi{B), such that, when Fi{D) operates upon y, and then F2,{D) operates upon the result of this operation, the same result is obtained as when F{D) operates upon y. This may be expressed symbolically, f{D)y = F,{D)\F,{D)y\; or simply, f{D)y = F,{D)F,{D)y, it being understood that the operations indicated in the second member of the last equation are made in order from right to left. Factors of this kind have already been employed in deal- ing with linear equations with constant coefficients, and with §88.] SOLUTION FOUND BY FACTORING. 113 the homogeneous linear equations, Arts. 63, 56, 67, etc. With the exception of the classes of equations just mentioned, the factors are generally not commutative ; this can be verified, in the case of the examples belovf. If one of the integrals be known, its corresponding factor is known, and the second factor can be determined by means of the equation and the known factor. For instance, if ?/ = e" be an integral of the given equation, then (Z) — l)y is the corre- . sponding factor ; it y = x he a.n integral, (xD — l)y is the corresponding factor. The following example will make the method of procedure clear. ^g + (-^)|- This equation, whicli is Ex. 1, Art. 87, when written in the symbolic form, is [_xD^ + (l~x)D-l]y = e''; (1) on using symbolic factors, it becomes (xD + l)(D--\)y = e-. (2) [These factors are not commutative, for {D — l)(xD + l)y on expan- sion gives {xD"' +(_2 -x)D- 1}!/]. Let (D - l)y = v, (3) and (2) becomes (xD + 1)« = e' ; whence, v = cx-i + e'x-'^. Substitution of this value of v in (3) gives (D — l)y = cx-^ + e^x-' ; whence, on integrating, y = cie'^ + ce' i e-^x-^dx + e' log a, the solution found in the last article. Ex. 2. Solve Ex. 3, Art. 87, by this method. Ex.3. Solve 3 x2g+ (2 -6*2) J- 42/ = 0. ^ Ex.4. Solve 3a;2^-|-(2 + 6a;-6x2)^-4!/ = 0. dal^ ^ dx 114 DIFFERENTIAL EQUATIONS. [Ch. IX. 89. Solution found by means of two first integrals. It fol- lows, from a statement made in Art. 75, that a linear equation of the second order has' two first integrals of the first order. If these integrals be known, then -^ can be eliminated be- tween them ; the relation thus found between x and y will be a solution of the original equation. Another method of solution that can be used in the case of > the linear equation of the second order is the '! method of variation of parameters."* As most of the equations solvable by it are solvable in other ways, and as it is rather long, it will not be given here. (See Johnson's Diff. Eq., Arts. 90, 91 ; Forsyth's Diff. Eq., Arts. 65-67.) Ex. Solve a^f -^1 =H-f^j by means of the first integrals. On putting ^=p, y^= j^i ^^^ integrating, there appears a first mtegral On substituting for ^ its equivalent expression p -~, and integrating, ctx ay another first integral is obtained, a2p2=(2/-|-C2)2-a2. The elimination of p between these first integrals gives the solution !/ + C2 = a cosh 90. Transformation of the equation by changing the dependent variable. Sometimes an equation can be transformed into an integrable type by changing the dependent variable. If any linear equation of the second order, * This method is due to Lagrange. § 89-91.] CHANGE OF THE DEPENDENT VARIABLE. 115 be taken, and y^v be substituted for y therein, y■^ being some function of x, (1) will be transformed into do?' which has v for its dependent variable. This equation may be written daf dx yi where Pi = P + -^, (4) 2/i dx -^ ^-^@+^S+e4 (5) Any value desired can be assigned to Pi or Qi by means of a proper choice of 2/i- Thus, Qi will be zero if 2/1 be chosen so this is what was done in Art. 85. Again, P„ the coefficient of the first derivative in (3), can have any arbitrary value assigned to it ; but then yi must be chosen so as to satisfy (4) ; that is, 2/1 = e*J'(^i--P''^- (6) 91. Removal of the first derivative. In particular, it follows from (4) or (6) Art. 90 that Pj is zero if 2/1 = e-iJ"^*^. On substituting this value of yi in the coefficient of v in (2) Art. 90, this coefficient becomes O — 1 '^^ — 1 P2 116 DIFFERENTIAL EQUATIONS. [Ch. IX. Therefore the differential equation (1) Art. 90 of the second order is transformed into a differential equation not containing the first derivative, by substituting and the transformed equation is g+e..=x, (1) where Qi=Q-i^-ip2, ^nd X^ = Xe^l''^. dx The new equation (1) may happen to be easily integrable. Transforming (1) Art. 90 into the form (1) is called " removing the first derivative." The student should memorise the above values of the new Qi and X,, in terms of P, Q, X, for then he can immediately write down the new equation in v, without the labour of making the substitution in the original equation and reducing. It may be remarked in passing that this removal of the term next to the second derivative is merely an example of the general theorem, that the coefiicient of the tejfm of (n — l)th order in a linear equation can be removed by substituting yiV for y, where --IPldx 2/1 = e "J The reader can easily verify this by making the substitution. (See Forsyth's Diff. Eq., Art. 42.) Ex.1. Solve^ + i-^+fJ L_6^j,^0. ^.juj; \^^j gjjj Here P = a;"i, Q = — ^ — ; and hence yi - e-J/™' = e"'^'. 4 X J 6 x^ * § 92.] CHANGE OF THE INDEPENDENT VARIABLE. 117 If the second term be removed, and hence the transformed equation is the solution of which is -0, v = eix' + ^- Hence the general solution of the given equation is Ex. 2. Solve 4 x^^ + 4 a;^^ + (a;8 + 6 «* + 4)?/ = 0. Ex. 3. Solve ^-2tanx^+52/ = 0. dz^ ax Ex.4. Solve a;2^-2(a;2 + a;)^ + (K2 + 2x + 2)2/ = 0. 92. Transformation of the equation by changing the independent variable. An equation can sometimes be transformed into an integrable form by changing the independent variable. Suppose that g+p|+Qy=X (1) is any linear equation of the second order, and that the inde- pendent variable is to be changed from x to 2, there being some given relation, z =f(x), connecting x and z. dy cPz dz daf dy dydz , d^y d^y/dzV ( Since -f- = :f:r-> and ■:A = :fi(:r] +" dx dz dx da? dz' \dxj ( (1) becomes g + P,|+Q^ = X>, (2) d'z , pdz where P,=^^, Q.= j§y, ^r., X,=j§-, (3) -7^^''^'- fdzY '-fdzY \dx) W Wa^y 118 DIFFEBENTIAL EQUATIONS. [Ch. IX. P], Qi, Xi, as just expressed, are functions of x; but can be immediately expressed as functions of z by means of the relar tion connecting z and x. Any arbitrary value can be given to P^; but then z must be so chosen that it satisfies the first of equations (3). In particular, Pj will be zero if ^+ P^= 0, that is, if » = C e-i'-'^'dx. ax' ax J Again, the new coefficient Q, will be a constant, a^, by virtue of the second of equations (3), if a^i — j = Q, that is, if a» = j VQ dx. Ex.1. Solve^' + ?^ + ^2, = 0. dx'^ X dx X* Find z, such that f ^V = ^ ; solving, z = ±-- \dxj X* X Change of the independent variable from x to a will now give and this has for its solution y = A cos 2 + B sin z. Hence the solution of the given equation is a . a y = C1COS-+ C2 sill-' Ex. 2. Solve -4 + cot X -f^ + 4 ?/ ooseo- x = 0. dx^ dx Ex.3. Solve X ^ - $^ + 4 x3(/ = x5. dx^ dx Ex. 4. Solve x6 '^ + 3 x^^ + a^y = A- fte^ dx X-' 93. Synopsis of methods of solving equations of the second order. This article is merely a synopsis of all the methods discussed thus far in the book that are employed in the solution of equay § 93.] SYNOPSIS OF METHODS OF SOLUTION. 119 tions of the second order. Several of these methods may be suitable for solving the same equation. The references are to the chapters and articles where the methods are described. The student is advised to select a few equations of the second order from the articles referred to, and' to solve each one in two or more different ways. An equation of the second order may be (a) linear with constant coefBcients, [Chap. VI.] (6) a homogeneous linear equation, [Chap. VII.] (c) an exact differential equation, [Arts. 73-75, 76] (d) an equation that does not directly contain the dependent variable, [Arts. 76, 78] ; (e) an equation that does not directly contain the independent variable, [Arts. 77, 79] ; (/) in the form ^ =/(.y), [Art. 77] ; (g) an equation, one of whose integrals is known or is easily found by inspection, [Arts. 85, 87] ; (h) factorable into symbolic operators, [Art. 88] ; (i) an equation of which two first integrals can be easily found, [Art. 89] ; (j) an equation that can be integrated in series. [Art. 82]. If the equation is not in an integrable form, it may be put in such a form by (a) so changing the dependent variable, that (1) the coefficient of the first derivative will have an assigned value [Art. 90] ; or that (2) (in particular), this coefficient will be zero [Art. 91] ; (6) so changing the independent variable, that (1) the equation will be transformed into the linear form with constant coefficients, or into the homogeneous linear form [Art. 71] I 120 piFFEBENTIAL EQUATIONS. [Ch. IX. or that (2) the coefficient of the first derivative will have an assigned value, and, in particular, the value zero [Art. 92] ; or that (3) the coefficient of the variable will have an as- signed value, aM, in particular, be a constant [Art. 92]. EXAMPLES ON CHAPTER IX. =■ S^IS-(»--S)'=»- 1 ^ + ?^-„2, 5. (a; - 3) -=-| - (4 a; - 9) -^ + 3(a; - 2)^ = 0, e" being a solution. 8. a; ^ — (2 X — 1) -^ + (a: — 1 )w = 0, given that w = e' is a solution. da;'' ■^dx ^ /» > o 9. (l-a;2)g + a:|-2, = a;(l-a;^)l. 10. ra;sina;+cosa;)^-xcosa;3^+2/cosa;=0, of which 3/= a; is a solution. ^ ' dx^ dx difi^ " dx^ ~ dx 12. (1 — ^^)-^ — x—-— a^y =0, of which y = ce"""''* is an integral. 14. x2^ _ 2x(l + x) f^ + 2(1 + x)y = x'. dx^ ^ dx ^ ^ dx^ X dx a § 94, 95.] GEOMETRICAL APPLICATIONS. 121 CHAPTER X. GEOMETRICAL AND PHYSICAL APPLICATIONS. 94. Chapter V. was devoted to geometrical and physical applications ; but the choice of problems for that chapter was restricted by the condition that a differential equation of an order higher than the first should not be needed in determin- ing their solution. The practical problems now to be givep are of the same general character as those already set ; but h order to obtain their solution, equations of orders higher than the first may be required. 95. Geometrical Problems. The following can be added to the geometrical principles and formulae given in Art. 42. The radius of curvature in rectangular co-ordinates is da? If the normal be always drawn towards the £B-axis, both it and the radius of curvature at any point on the curve are d''v drawn in the same direction when y and —4 at the point are dur opposite in sign, and they are drawn in opposite directions when y and —^ agree in sign. This will be apparent on draw- ing four curves, one concave upward and one concave down- ward, above the a^axis, and two similar ones below this axis. 122 DIFFERENTIAL EQUATIONS. [Ch. X. Ex. Find the equation of the curve for any point of which the second derivative of the ordinate is Inversely proportional to the semi-cubical power of the product of the sum and difference of the abscissa and a con- stant length a ; determine the curve so that it will cut the ?/-axis at right angles, and the K-axis at a distance a from the origin. The first condition is expressed by either of the equations cPy ^ Ifi ^ and 2 + i2)« = ; * A particular choice of units is presupposed in this statement. 124 DIFFERENTIAL EQUATIONS. [Ch. X. hence s = ci sin Ai + ca cos kt ; that is, s = a sin (kt + 6), as above. Ex. 2. In the case of the simple pendulum of length I, the equation connecting the acceleration due to gravity and the angle 6 through which the pendulum swings Is when is small. Determine the time of an oscillation. Since ^ + 29 = 0, = cx cos -v/^J + C2 sin -v/^<. I when J = ; applying the =:0ts cos yi^-t; that is, t =-V- cos-i— , the time of a complfete oscillation from do to — 0o and back again is 2 ir -v— ' fi Let = 0a and — = when t = 0; applying these conditions, Ci = 00, C2 = 0, and hence ) = fln fins \ . which is the time of swing from 0o to 0. It 9 = —0o, t = ir-v/i; Jience EXAMPLES ON CHAPTER X. 1. Determine the curve in which the curvature is constant and equal to k. 2. Determine the curve whose radius of curvature is equal to the normal and in the opposite direction. 3. Determine the curve whose radius of curvature is equal to the normal and in the same direction. 4. Determine the curve whose radius of curvature is equal to twice the normal and in the opposite direction. 5. Find the curve whose radius of curvature is double the normal and in the same direction. 6. Determine the curve whose radius of curvature varies as the cube of the normal. § 96.] EXAMPLES. 125 7. Find the curve whose radius of curvature varies inversely as the abscissa. 8. rind tlie distance passed over by a moving particle when its acceleration is directly proportional to its distance from a fixed point, the acceleration being directed away from the point from which distance is measured. 9. Find the distance passed over by a particle whose acceleration is constant and equal to a, vo being the initial velocity, and so the initial distance of the particle from the point whence distance is measured. 10. Find the distance passed over by a particle when the acceleration is inversely proportional to the square of the distance from a fixed point. 11. Find the distance passed over by a body falling from rest, assum- ing that the resistance of the air is proportional to the square of the velocity. 12. The acceleration of a moving particle being proportional to the cube of the velocity and negative, find the distance passed over in time t, the initial velocity being vq, and the distance being measured from the position of the particle at the time J = 0. 13. The relation between the small horizontal defl.ection 9 of a bar magnet under the action of the earth's magnetic field is A — +MI£e = 0, where A is the moment of inertia of the magnet about the axis, M the magnetic moment of the magnet, and H the horizontal component of the intensity of the field due to the earth. Find the time of a complete vibration. 14. In the case of a stretched elastic string, which has one end fixed and a particle of mass m attached to the other end, the equation of motion is where I is the natural length of the string, and e its elongation due to a weight mg. Find s and v, determining the constants so that s = so at the time { = 0. 15. A particle moves in a straight line under the action of an attrac- tion varying inversely as the (f)th power of the distance. Show that the velocity acquired by falling from an infinite distance to a distance a from the centre is equal to the velocity which would be acquired in moving from rest at a distance a to a distance -• 4 126 DIFFERENTIAL EQUATIONS. ' [Ch. X. 16. A particle moves in a straight line from rest at a distance a towards a centre of attraction, the attraction varying inversely as the cube of the distance. Find the whole time of motion. 17. The differential equation for a circuit containing resistance, self- induction, and capacity, in terms of the current and the time, is dp Ldt LG~ L^ ^'^' /(J) being the electromotive force. Find the current i. 18. The differential equation for the above circuit in terms of the charge of electricity in the condenser is dt^ Ldt LC U^'' Find the charge q. 19. Solve ^ + ^^+-^=0 when iJ20= 4 i. d«2 Ldt LC di I "^* 20. Solve L — 1- = 0, the differential equation which means that dt C ^ the self-induction and capacity in a circuit neutralize each other. Deter- mine the constants in such' a way that I is the maximum current, and i = when « = 0. (The given equation, on differentiation, reduces to — --\ = 0.) dt" juO 21. When the galvanometer is damped, the equation of motion may be written dP dt ^ ' a. being the deflection of the needle from the position from which angles are measured, when in its position of equilibrium, the factor k depending on the damping, and w^ on the restoring couple. Find the position of the needle at any instant. (Emtage, Electricity and Magnetism, pp. 179, 180.) * 22. Find the equation of the elastic curve for a cantilever beam of uniform cross-section and length I, with a load P at the free end, the differential equation being where / is the moment of inertia of the cross-section with respect to the * Merriman, Mechanics of Materials, pp. 72, 73. § 96.] EXAMPLES. 127 neutral axis, and E is the coefficient of elasticity of the material of the beam. (The origin being taken at the free end of the beam, the a;-axis being along its horizontal projection, and the t/-axis being the vertical, S = when x = l, and w = when a; = 0. These conditions are suffi- dx oient to determine the constants.) * 23. Find the elastic curve when the load is uniformly distributed over the beam described in Ex. 22, say w per linear unit, the differential equation being ^^d^^ — T- t 24. Find the elastic curve for the beam considered in Ex. 23, when a horizontal tensile force Q is applied at the free end, the differential equa- tion being * Merriman, Mechanics of Materials, pp. 72, 73. t Merriman and Woodward, Higher Mathematics, Prob. 106, p. 153. 128 DIFFERENTIAL EQUATIONS. [Ch. XI. CHAPTER XI. ORDINARY DIFFERENTIAL EQUATIONS "WITH MORE THAN TWO VARIABLES. 97. So far equations containing two variables have been con- sidered. It is now necessary to treat a few forms containing more than two variables. Such equations are either ordinary OT partial, the former having only one independent variable, and the latter more than one. In this chapter ordinary differ- ential equations will be discussed. 98. Simultaneous differential equations which are linear. First will be considered the case in which there is a set of relations consisting of as many simultaneous equations as there are dependent variables; moreover, all the equations are to be linear. By following a method somewhat analogous to that employed in solving sets of simultaneous algebraic equations that involve several unknowns, the dependent variables corresponding to the unknowns, there is obtained, by a process of elimination, an equation that involves only one dependent variable with the independent variable ; and from this newly formed equation an integral relation between these two variables may be derived. Then a relation between a second dependent variable and the independent variable can be deduced, either (1) by the method of elimination and integration employed in the case of the first variable; or (2) by substituting the value already found for the first variable, in one of the equations involving only the first and second dependent variables and the independent variable. The complete solution consists of as many indepen- §97,98.] SIMULTANEOUS LINEAR EQUATIONS. 129 dent relations between the variables as there are dependent variables. The following example will make the process clear : Ex. 1. Solve the simultaneous equations, ^ ^ clt dt " (2) ^ + 5x + 32/ = 0. Diflerentiation of (2) gives (3) ^+5^+3^ = 0. ^ ^ a{2 ^ dt dt These three equations suf&oe for the elimination of x and — ; this elimination is effected by multiplying the first equation by — 5, the second by 2, the third by 1, and adding ; the result is Solving (4), y = A cos t -\- Bsint; and substituting this value of y in (2), ^=_M±^cos«+A^l^sin«. 5 5 By using the symbol D, which was employed in Chap. VI., the elimina- tion can be effected more easily. On substituting D for — , the given equations become (i) + 2)x + (I> + l)2/ = 0, 5x+(i> + 3)^ = 0. Eliminating x as if D were an algebraic multiplier, (i)2 + i)y = 0, which is equation (4); the remainder of the work is as above. If y had been eliminated instead of x, the resulting equation would have been (Z)2 + l)x = ; whence x = A' co&t+ B' sin { ; 130 diffebBjsttial equations. [Ch. XI. substitution of tliis value in dx (5) ^-Sx-2y = 0, at which is (1) minus (2), gives „ _ SB' + A' sin t + — — ^' cos t. Substitution is made in (5), because it is easier to derive the value of y from it than from (1) or (2). The second form of solution comes from the first on substituting A' for - h^L+Ji and B' for ^^ ~ ^ ^ the coefficients in the first value of 5 ' 5 X. In general the constants are arbitrary in the value of only one of the dependent variables. Ex. 2. Solve dx dt 7a;+ 2/ = ^-2a;-52/ = dt Ex.3. Solve —-\-2x~Sy = t dt dy dt 3x + 2y = e^ Ex.4. Solve4^ + 9^+44x + 49j/ = « dt dt 3^ + T^+3ix + 3Sy = e' dt dt " Ex.6. Solve ^-3x-4j/ = dt'' 11+ x+ . = 99. Simultaneous equations of the first order. Simultaneous equations of the first order and of the first degree in the derivatives can sometimes be solved by the following method, which is generally shorter than that shown in the last article. Equations involving only three variables will be considered; the method, however, is general, and can be applied to equa«- tions having any number of variables. § 99.] EQUATIONS OF FIRST ORDER AND DEGREE. 131 The general type of a set of simultaneous equations of the first order between three variables is ci-i '- (a, b) = c, or <^ {u, v) = 0, which is equally general, since c can fee involved in the arbi- trary function. This can be verified directly. Hence, if M = a, V = b, be independent integrals of the system (2) Art. 99, fj>(u,v) = is the general expression for the integrals of these equations. The arbitrary functional relation may just as well be written in the form u =f(v). This d eduction will be used in Art. 115. 101. Geometrical meaning of simultaneous differential equations of the first order and the first degree involving three variables. § 101.] GEOMETRICAL MEANING. 135 Equations (1) or (2) Art. 99 will determine, for each point d% d\i (x, y, z), definite values of — and ~ ; that is, these differential equations determine a particular direction at each point in space. Therefore, if a point moves, so that at any moment the co-ordinates of its position and the direction cosines of its line of motion (these cosines being proportional to dx, dy, dz, and hence to P, Q, R, by (2) Art. 99) satisfy the differential equa- tions, then this point must pass through each position in a particular direction. Suppose that a moving point P starts at any point and moves in the direction determined for this point by the differential equations to a second point at an infinitesi- mal distance ; thence, under the same conditions to a third point ; thence to a fourth point, and so on; then P will describe a curve in space, whose direction at any one of its points and the co-ordinates of this point will satisfy the given differential equations. If P start from another point, not on the last curve, it will describe another curve ; through every point of space there will thus pass a definite curve, whose equation satisfies the given differential equations. These curves are the intersections of the two surfaces which are represented by the two equation's forming the solutions ; for, these two equa- tions together determine the points and the ratios of dx, dy, dz, thereat which satisfy the differential equations. Moreover, the curves are doubly infinite in number; for they are the intersections of the surfaces represented by the independent integrals u = a, v = b, and each of these equations contains an arbitrary constant which can take an infinite number of values. Thus, the locus of the points that satisfy the differential equations of Ex. 1, Art. 99, is the curves, doubly infinite in number, which are the intersections of the system of planes whose equation is y = az, with the system of spheres whose equation is . a;^ + 2/' + 2^ = bz; 136 BIFFEBENTIAL EQUATIONS. [Ch. XI. and the locus of the points that satisfy the equations of Ex. 2, Art. 99, is the curves which are the intersections of all the planes represented by Ix + my + n« = c, c having an infinite number of values, with all the spheres «? +y^ + z' = k% k having an infinite number of values. 102. Single differential equations that are integrable. Con- dition of integrability. The equation Pdx + Qdy + Edz = (1) has an integral u = a, (2) when there is a function u whose total differential du is equal to the first member of (1), or to that member multiplied by a factor. If (1) have an integral (2), then, since du = — dx + —dy + --dz, ax ay dz P, Q, B, must be proportional to — , — , — : that is, ^ ^ dx dy dz ' p du These three conditions can be reduced to one involving the coefficients P, Q, R, and their derivatives. On differentiating the first of these three equations with respect to y and z, the second with respect to z and x, and the third with respect to X and y, there results, § 102, 103.] 8IN0LE INTEGSABLE EQUATION. 137 6y dy 5a; dx 8\ dxdy _ d^u _ dydz dzdx dx ax dy ^ 8y dz dz whence, on rearranging, comes fdP fdR 3Q fla; dy dP dz oa; dy - K>^_0^ = i? dy Q :ifl_:ii_] = p^f^_ii dz ,dfi dx On multiplying the first of the last three equations by E, the second by P, the third by Q, and adding, there is obtained, = 0; whence i«')<, (a + mza') (x + miy) + c + m^c' = A2e'-'+'^'"i' , where mi, toj, are the roots of a'm^ + (a - b')m -6=0; and obtain a similar solution for the system ^=ax + by, '^ = a'x + b-y. dt^ at' (Johnson, Dig. Eq., Ex. 16, p. 269.) 18. Find the equation of the path described by a particle subject only to the action of gravity, after being projected with an initial velocity «„ in a direction inclined at an angle to the horizon. 19. Determine the path of a projectile in a resisting medium such as air when the retardation is c times the velocity, given that the initial velocity is v^ in a direction inclined at an angle 4> to the horizon. 20. F'nd the path described by a particle acted upon by a central force, the force being directly proportional to the distance of the particle. 21. The two fundamental equations of the simple analytical theory of the transformer are iJiii + ii^+ilff = «i, dt dt dt dt where ii, I'a, denote the currents, Mi, R2, the resistances, Li, Li, the coefficients of self-induction of the primary and secondary currents re- spectively, ei the impressed primary electromotive force, and M the mutual induction. § 106.] EXAMPLES. 145 Show that, e, ii, i^, and t being variable, the differential equations for the primary and secondary currents respectively are, (iii2 -M-^)^ + (L1B2 + L2B1) ^ + BiB^h = B,e, + ij^, at'' ox cit dfi dt dt (Bedell, The Principles of the Transformer, Chap. VI.) 22. The general equations for electromotive forces iii the two circuits of a transformer with capacities Ci and ca being ci dt at oJ i2dt ' J . J . Ca dt dt where e, ii, h, t, are variable, show that the differential equations for the primary and secondary currents are (iii2 -M-')^ + (BrL, + B,LO ^+(1^+^+ B,B,)^ dt* dt^ \ C2 ci / dt^ "it). \C2 ci j at C1C2 C2 (Lii2 - M-^) ^ + iB,U + B,L,) ^ + (Ll + ^+ ^i-R^VIi dt* dt^ \ C2 ci / dt^ + (^ + ^]^ + J-ir, = -Mf'"{t-). \ Cj Ci / df C1C2 (Bedell, The Principles of the Transformer, Chap. XI.) 146 DIFFERENTIAL EQUATIONS. [Ca. XII, CHAPTER XII. PARTIAL DIFFERENTIAL EQUATIONS. 107. Definitions. Partial differential equations are those which contain one or more partial derivatives, and must, there- fore, be concerned -with at least two independent variables.* The derivation of partial differential equations will be dis- cussed in Arts. 108, 109 ; equations of the first order will be considered in Arts. 110-123; and those of the second and higher orders in the remaining part of the chapter. These equations, excepting the ones treated in Arts. 117, 134-136, will involve only three variables. In what follows, x and y will usually be taken as the independent variables, and z as dependent; the partial differential coefficients — , — , will be denoted by p and q respectively. ^ 108. Derivation of a partial differential equation by the elimi- nation of constants. Partial differential equations can be derived in two ways : (a) by the elimination of arbitrary con- stants from a relation between x, y, z, and (6) by the elimination of arbitrary functions of these variables. To illustrate (a) take ^ {x, y, z, a, 6) = (1) a relation between x, y, z, the latter variable being dependent upon x and y. In order to eliminate the two constants a, 6, * Equations with partial derivatives were at first studied by D' Alembert (sea p. 173), and Euler (see p. 64), in connection witli problems of physics. §107,108.] DURIVATION. 147 two more eqiiations are required. These equations can be obtained from (1) by differentiation with respect to x and y ; they will be 6a; dz dy dz By means of these three equations, a and b can be eliminated, and there will appear a relation of the form F(x,yhP,q)=0, (2) a partial diiferential equation of the first order. In (1) the number of constants eliminated is just equal to the number of independent variables, and an equation of the first order arises. If the number of constants to be eliminated is greater than the number of independent variables, equations of the second and higher orders will, in general, be derived. The following examples will illustrate this. In these exam- ples, z is to be taken as the dependent variable. Ex. 1. rorm a partial differential equation by the elimination of the constants h and k from (x - hy +{y- ky + z^ = o\ Differentiating with respect to x and y, x — h + zp = 0, ' y -k + zq = 0. Substituting the values ol x — h, y — k from the last two equations m the given equation, a2(p2 + g2 4. 1) := c2. Ex. 2. Form the partial differential equation corresponding to z = az + hy + ab. Ex. 3. Eliminate a and 6 from z = a{x + 2/) + 6- Ex. 4. Eliminate a and h from z = ax ■\- a'^y^ + b. Ex. 5 Eliminate a and b from z =(x + a)(y + b). Ex. 6. Form a partial differential equation by eliminating a, b, c from T.2 yi g2 5_ + lL + £, = l. 148 DIFFERENTIAL EQUATIONS. [Ch. XII, 109. Derivation of a partial differential equation by the elimina- tion of an arbitrary function. To illustrate (b) of Art. 108, sup- pose that u and v are functions of x, y, z, and that there is a relation between u and v of the form is arbitrary. The relation may also be expressed in the form u =f(v), where / is arbitrary. It is now to be shown, that, on the elimination of the arbitrary function <^ from (1), a partial differential equation will be formed ; and, moreover, that this equation will be linear, that is, it will be of the first degree in p and q. Differentiation of (1) with respect to each of the independent variables x and y gives ■ f:(l;+-l)+s*(l+-lf)-''' l*(g+'l)+l*(|+'l;)=»- Elimination of -5, -^, from these two equations results in au av and this can be rearranged in the form Pp+Q.q = E, (2) where p^dudv_dudv dy dz dz By ■ Q _ flw 9« _ 9m dv dz dx dx dz' j^ _dudv^_Su dv dx dy dy dx Thus, from (1), which involves an arbitrary function <^, a par- tial differential equation (2) has been obtained, which does not contain '{x — y). Differentiating with respect to y, q = ne'^y{x — y)—e''y{x'^ -\- y^ + z"^). Ex. 4. Eliminate the function / from z = y'^ + 2/( - -)- log j/ ) ■ Ex. 5. Eliminate the arbitrary functions / and from z =f(x + ay)+{x - ay). Partial Diffekential Equations of the First Order. 110. The integrals of the non-linear equation: the complete and particular integrals. In Art. 108 it was shown how the partial differential equation F(x,y,z,p,q) = (1) may be derived from ^ («, y, 2, a, b) = 0. (2) Suppose, now, that (2) has been derived from (1), by one of the methods hereafter shown ; then the solution (2), which has as many arbitrary constants as there are independent variables, is called the complete integral of (1). A particular integral of (1) is obtained by giving particular values to a and b in (2). * See Edwards, Differential Calculus, Arts. 509-514 ; Williamson, Differential Calculus, Arts. 315-319 ; Johnson, Differential Equations, Arts. 299-301. 150 DIFFERENTIAL EQUATIONS. [Ch. XII. 111. The singular integral. The locus of all the points whose co-ordinates with the corresponding values of p and q satisfy (1) Art. 110, is the doubly infinite system of surfaces represented by (2). The system is doubly infinite, because there are two constants, a and b, each of which can take an infinite number of values. Since the envelope of all the sur- faces represented by <^ (x, y, z, a, 6) = is touched at each of its points by some one of these surfaces, the co-ordinates of any point on the envelope with the p and the q belonging to the envelope at that point must satisfy (1) ; and, therefore, the equation of the envelope is an integral of (1). The equa- tion of the envelope of the surfaces represented by (2) is obtained in the following way : * Eliminate a and h between the three equations, ^ (», y, z, a, h) = 0, da db ' and the relation thus found between x, y, z is the equation of the envelope. This relation is called the singular integral ; it differs from a particular integral in that it is not contained in the complete integral ; that is, it is not obtained from the com- plete integral by giving particular values to the constants. (Compare Arts. 32, 33.) 112. The general integral. Suppose that in (2) Art. 110, one of the constants is a function of the other, say b =/(a), then this equation becomes {x,y,z,a,f(a)) = 0, (1) which represents one of the families of surfaces included in the system represented by (2). The equation of the envelope *T"or proof see C. Smith, Solid Geometry, Arts. 211-215; W. S. Aldis, Solid Geometry, Chap. X. § 111, 112.] TBE GENERAL INTEGRAL. 161 of the family of surfaces represented by (1) will also satisfy (1) Art. 110, for reasons similar to those given in the case of the singular integral. Moreover, this equation will be different from that of the envelope of all the surfaces, and it is not a particular integral. It is called the general integral ; and it is found by eliminating a between ^ («, y, z, a, f{a)) = 0, and ^=0. da These two equations together represent a curve, namely, the curve of intersection of two consecutive surfaces of the system (x, y, e, a, f(a)) = 0. The envelope of the family of surfaces, being the locus of the ultimate intersections of the surfaces belonging to the family, that is, of the intersections of consecu- tive surfaces, contains this curve to which the name charac- teristic of the envelope has been given. Hence the general integral may be defined as the locus of the characteristics. Other relations may appear in the process of deriving the singular and the general integrals from the complete integral, but it is beyond the scope of this work to discuss such relations. When one has performed the operations necessary to find the singular and the general integrals, he should test his result by trying whether it satisfies the differential equation. (Compare Arts. 33-38.) In the case of every equation, the general integral and the singular integral, as well as the complete integral, must be indicated or the equation is not considered to be fully solved. The complete integral is to be found first, and from it the other two are to be derived.* It is evident that the locus of the singular integral will be the envelope of the loci of all the other integrals, of the general as well as of the complete. * The distinction between the three kinds of integrals of partial dif- ferential equations was made by Lagrange in Memoirs of the Berlin Academy, 1772, 1774. 152 DIFFERENTIAL EQUATIONS. [Ch. XII. Ex. In Ex. 1, Art. 108, the differential equation 2=(p2 + 9^ + l)=c2 (1) was derived from {X - ny + (2/ -ky + z^ = c^. (2) The latter equation, which contains two arbitrary constants, is the complete integral of the former ; it represents the doubly infinite system of splieres of radius c, whose centres are in the xy plane. A particular integral of (1) is obtained by giving ft and h particular values in (2) ; thug, (a; - 2)2 +(2/- 3)2 + s2 = c2 is a particular integral. The singular integral of (1) is the equation that represents the envelope of these spheres ; \t is obtained by eliminating ft and k from (2) by means of the relations derived by differentiating (2) with respect to ft and k. The differentiation gives K — ft = 0, y-k = Q; on substituting these values in (2), ft and k are eliminated, and there results the equation 3 = ± c. This satisfies equation (1), and, therefore, is the singular integral. It represents the two planes that are touched by all the spheres represented by (2). Suppose, now, that one of the constants is made a function of the other, say, that k = h. Then the centres, since their co-ordinates have that relation, are re- stricted to the straight line t/ = x in the xy plane ; and of the system of spheres representing (2) there will be chosen a particular family, namely, {X - ft)2 + {y- ft)2 -f ^2 _ c2. (3) The envelope of this family is the tubular surface, in this case a cylin- der, which is generated by a sphere of radius c, when its centre moves along the line y = x. The equation of this envelope is a general integral ; it is found by eliminating ft from (3) by means of the relation obtained by differentiating (3) with respect to ft.. The differentiation gives x — ft-l-j/ — ft = 0, whence ft = J(x -f y). § 113.] INTEGRAL OF THE LINEAR EQUATION. 153 Substituting this value of h in (3), x2 + 2/^ _ 2xs/ + 2 22 = 2 0*2, which is a general integral. If the relation between the constants were assumed to be fc2 z= 4 ah, the corresponding general integral would be the equation of the tubular surface generated by a sphere of radius c, whose centre moves along the parabola y^ = iax in the xy plane. 113. The integral of the linear equation.* In Art. 109 it was shown that from an arbitrary functional relation {u,v) = Q (1) there is derived, by the elimination of the function ^, a linear partial differential equation Pp+Qq = R. (2) Suppose that (1) has been derived from (2) ; then (j>(u, «) = is called the general solution of (2). Since <^ is an arbitrary function, the solution (1) is more general than another solution of (2) that merely contains arbitrary constants. For instance, Ex. 2, Art. 109, shows that the general solution of yp — xq = is z = F{a? + 2/2), where F denotes an arbitrary function. The arbitrary func- tion F may take various forms, as, z = a{o? + yy+b{3? + y^, z = a sin (a^ + 2/^ + h, etc., which are all solutions of the differential equation, and are included in the general solution above. *The student will find it of great advantage to read C. Smith, Solid Geometry, Arts. 216-226 ; W. S. Aldis, Solid Geometry, Arts. 142-151, in connection with this and following articles. 154 DIFFEBENTIAL EQUATIONS. [Ch. XII. 114. Equation equivalent to the linear equation. The type of a partial differential equation which is linear in p and q is Pp+Qq = B, (1) JP, Q, R being functions, of x, y, z. Suppose that u = a is any relation that satisfies (1); differentiation with respect to X and y gives fin , du to + 8-z^- = 0, flu , du + -z-Q = :0; dy dz^ du to ( &z ( du whence P = — ^, ^ = — -r^- du d% Substitution of these values of p and q in (1) changes it to p|^ + Q|^+i?|-" = o. (2) to dy dz Therefore, if m = a be an integral of (1), u = a also satisfies (2). Conversely, if m = a be an integral of (2), it is also an integral of (1). This can be seen by dividing by — and substi- tuting p and q for the values above. Therefore equation (2) can be taken as equivalent to equation (1). 115. Lagrange's solution of the linear equation. In Art. 100, it was shown that is dudv_du dv\ , /' du dv_du dv\ _dudv _ du dv dy dz dz dy) \dz dx dx dz J dx dy dy dx Comparison shows that these equations have the forms dx _dy _ dz and Pp-\- Qq = B,, respectively. Ex. 1. Solve xzp + yzq = xy. Dividing by xyz, " + ? = -; y X z forming the auxiliary equations, ydx = xdy = zdz. Integrating the equation formed by the first two terms, X I Also y(lx + xdy = 2zdz ; whence z^ — xy = c. Therefore, the solution is z^ — xy = l-\ , oi f I z^ — xy, - ) = 0. Ex. 2. Solve p + q = -. a Ex. 3. Solve (mz — ny)p + (_nx — lz)q = ly — mx. Ex. 4. Solve x^p + y^q = z^. Ex. 5. Solve ^ + xzq = y"^. 117. The linear equation involving more than two independent variables. If there be n functions u-^, u^, ■■■, u„, of m + 1 varia- bles z, ojj, Xi, •••, x„, z being dependent and the other variables § 117.] MORE THAN TWO INDEPENDENT VARIABLES. 157 independent, then the arbitrary function <^ can be eliminated from <^(Mi, Mj, ..., M„)=0 (1) by an extension of the method used in Art. 109. The result will be a linear partial differential equation of the form P.^ + P.^+-+P„P- = B. (2) Moreover, on forming the differential equations correspond- ing to Ml = Ci, Mj = Cj, • • •, M„ = c„, by eliminating the constants c„ C2, •••, c„, and proceeding as in Art. 116, there will be obtained TrT, — TrR ^^^ Hence the following rule may be given : In order to deduce the general integral of the partial differ- ential equation (2), write down the auxiliary equations (3), and find n independent integrals of this system of equations ; let these integrals be Mj = Ci, M2 ^ C2, • • •, M„ = c„ ; then ^(mj, u^, •■•, m„) = 0, where denotes an arbitrary function is the integral of the given equation. Suppose that m = c is an integral of (2) ; then du equation (2) can take the equivalent form P,|^ + A|^+-+P„|^ + i2^ = 0. (4) ox^ . 0x2 ox^ OZ 158 DIFFERENTIAL EQUATIONS. [Ch. XII. Ex.1. Solve (t + y+z)Si + (_t+x+z)^ + {t+x+y)^=x+y+z. dx dy 02 The auxiliary equations are dt dx dy _ dz x-\-y + z y + z + t z + t + x t + x + y whence, dt + dx + dy+dz _ dt — dx _ 3(^t + X + y + z) ~ X- t from this, log ({ + a; + 1/ + s)^ = log — 5J— ; X — t • 1 hence,- (x — t)(t + x + y + z)^ = ci ; similarly, (.y - t)(t + x -}- y + z)'s = ca, and (z — t)(t + x + y + z)^ = cs. Hence the solution is {(x — t)u, {y — t)u, (z — «)»} = 0, ■where it = (i + x + j/ + 2)1. Ex. 2. Solve x^ + y^+z^ = xyz. dx dy dz 118. Geometrical meaning of the linear partial differential equation. In Art. 105 it was shown that the curves whose equations are integrals of dx _dy._dz /-in 'p~'q~b ^^ are at right angles to the system of surfaces whose equation satisfies Pdx + Qdy + Rdz = 0. (2) Suppose that u = a, v = b are any pair of independent integrals of (1). Let a take a par- ticular value, say a^. The surface represented by m = aj is intersected by the system of surfaces whose equation is « = 6, in an infinite number of curves, a curve for each one of the infinite number of values that b can have. Thus u = ai repre- §118,119.] SPECIAL METHODS OF SOLUTION. 159 sents a locus which passes through, or upon which lie, curves infinite in number, that are orthogonal to the surfaces repre- sented by (2).t Therefore, since the general integral of Pp+Qq=B (3) is an arbitrary function of integrals of equations (1), any integral of (3) passes through a system of lines that are orthogonal to the surfaces forming the locus of (2) ; and hence the surfaces represented by (3) are orthogonal to the surfaces represented by (2). * 119. Special methods of solution applicable to certain standard forms. There are a few standard forms to which many equa- tions are reducible, and which can be integrated by methods that are sometimes shorter than the general method which will be shown in Art. 123. These forms will now be discussed. Standard I. To this standard belong equations that involve p and q only ; they have the form F{p,q)=0. (1) A solution of this is evidently z = ax + by + c, if a and 6 be such that F{a, 6) = ; that is, solving the last equation for b, if b =f(a). The complete integral then is z = ax + yf(a) + c. (2) The general integral is obtained by putting c = (a), where denotes an arbitrary function, and eliminating a between z = ax + yf(a) + ^ (a), and = a;-|-2//'(a)-h^'(«)- * Arts. 119-122 closely follow Forsyth, Differential Equations, Arts. 191-196. t When such surfaces exist. See Arts. 104-106. 160 DIFFERENTIAL EQUATIONS. [Ch. XII. .< The singular integral is obtained by eliminating a and c be- tween the complete integral (2) and the equations formed by- differentiating (2) with respect to a and c ; that is, between z = ax + yf{a) + c, = x + yf{a), = 1; the last equation shows that there is no singular integral. Ex. 1. Solve (1) p^^ + q'^ = mK The solution is z = ax -\- hy + c, if a^ + 6^ = rrfl. Therefore, the complete solution is (2) z = ax ■\- Vm^ — a:^y + c. To find the general integral, put c =f(a); then z = ax + Vm'^ — a^ y + /(a) ; differentiate with respect to a, o = x- " y + /(«); V ?)i^ — a^ and eliminate a by means of these two equations. A developable surface is the envelope of a plane whose equation con- tains only one variable parameter.* Therefore, the general integral in this case represents a developable surface. In particular, if c or /(a) be chosen equal to zero, then the result obtained by eliminating a is (3) ^2 = m\x^ -t- 2/2). The complete integral (2) represents a doubly infinite system of planes ; the particular integral obtained by putting c equal to zero represents a singly infinite system of planes passing through the origin ; and the gen- eral integral (3) represents the cone which is the envelope of the latter system of planes. Ex. 3. Solve (1) xV -l- y'^q'^ = z"^. This may be written f^V-^ [yMV = l. Put ^ = dX, ^= d)', \zQxl \zdyl X y dz — = dZ ; whence X = log x, T= log y, Z = \ogz; the equation then z * See C. Smith, Solid Geometry, Art. 221. §120.] STANDARD II. 161 '^ (If)"- (!!)■=■• which comes under Standard I. From the preceding example the complete integral is Z = aX+ Vl - «^ Y + log c ; hence, z = cx^"^^ "-, which is the complete integral of (1). The singular integral is z = ; the general integral is to be found in the usual way. Ex.3. Solve3p2_2g2 = 4pg'. Ex. 4. Solve 5 = 6". Ex. 5. Solve pq = Tc. 120. Standard II. To this standard belong equations analogous to Clairaut's; they have the form z=px + qy+f{p,q). (1) That the solution is z = ax + hy + f{a, h) (2) can easily be verified. This is the complete integral, since it contains two arbitrary constants. It represents a doubly infinite system of planes. In order to obtain the general integral, put b = <^(a), where ^ denotes an arbitrary function ; then z = ax + y4>{a) +f\a, 4>{a)\; differentiate this with respect to a, = a; + 2/<^'(a) +/'(«), and eliminate a between these equations. , In order to obtain the singular integral, differentiate z — ax + by+f{a, b) with respect to a and b, thereby getting the equations da db and eliminate a and b between these three equations. 162 DIFFERENTIAL EQUATIONS. [Ch. XII. Ex.. 1. Solve z =px + qy + pq. The complete integral is z = ax + by + ab. In order to find the singular integral, differentiate with respect to a and b ; this gives = X + 6, = y -i- a; elimination of a and b by means of these equations gives s = —xy. The general integral is the a eliminant of z = ax + yf{a) + af{a), = X + J//' (a) + «/'(a) +/(a), where / denotes an arbitrary function. Ex. 2. Solve B =px + qy — iy/pq. 121. Standard III. To this standard belong equations that do not contain x ov y \ they have the form F{z,p,q)=Q. (1) Put X for aj + ay, where a is an arbitrary constant, and assume z=f{x + ay)=f{X) for a trial solution ; then _ dz dX _ dz dz dX __ dz ^~dX'dx~dX' dX'dy~ dX' Substitution in (1) gives F(z,-^,a^) = 0, (2) V dX' dXj ' ^ ' which is an ordinary differential equation of the first order. The solution of (2) gives an expression of the form whence, — = dX ; <^ (z, a) integrating, /(«, a) = X + 6, §121.] STANDARD III. 163 and hence, x + ay + b=f(z, a) is the complete integral. The general and the singular integrals are to be found as before. This method of solving equations of Standard III. can be formulated in the following rule: dz Substitute ap for q, and change p to , X being equal to dX x+ ay; then solve the resulting ordinary differential equation between z and X. Ex. 1. Solve (1) ^2(^2 + g,2 _,. X) = cK dz On putting ap for q, changingp to , and separating the variables, (1) becomes zdz Va2 + 1 = dX. Integrating, - Va^ + 1 Vc^ - z^ = X+ 6 ; squaring, and substituting for X its value x + ay, (2) (a2 + i)(c2-z2)^(x + ay + 6)2. This is the complete integral of (1), since it contains two independent arbitrary constants a and 6. Differentiate (2) with respect to a and b, and eliminate a and 6 ; there results z2 = c2, which satisfies (1), and is thus the singular solution. In order to find a general integral, substitute for 6 some function of a, and eliminate a from the equation. In particular, on putting b =— ak — h, (2) becomes (3) (a^ + l)(G''-zi) = {x-h + a(y~k)f. (3) Differentiation with respect to a gives the equation 2 a(c2 - 22) = 2(2/ - k){x - h + a(y - k)}, which in virtue of (3) can be put in the form (4) ix-h + a(y-k)}{a{x-h:)-iy-k)] = 0. (4) 164 DIFFERENTIAL EQUATIONS. [Ch. XII. On eliminating a from (3) by means of the first component equation of (4), tliere appears the equation g2 _ c2 ; and on eliminating a by means of the second component equation, there comes (a; - hy + (,y- ky + e^ = c\ (5) The general integral is thus made up of tlie last two equatioiis, which represent two parallel planes and a sphere. The planes and sphere form the envelope of the cylinders represented by (3). Equation (5) may also be regarded as a complete integral, if h and k be taken as arbitrary con- stants. (See Ex. 1, Art. 108 and Art. 112.) Ex.2. Solve q^y^ = z{z—px). This may be written and putting dYtov ^, t^Xfor — , (whence F= log?/ and X= log*), y X, the latter equation becomes which belongs to Standard III. Ex. 3. Solve 9(p% + ^a-j _ 4, Ex. 4. Solve p(_l + g-2) = g(s - a). Ex. 5. Solve pz=l + q^. 122. Standard IV. To this standard belong eqiiations that have the form f^{x, p) =f^{y, q). (1) In some partial differential equations in which the variable z does not appear, it happens that the terms containing p and X can be separated from those containing q and y ; the equation then has the form (1). Put each of these equal expressions equal to an arbitrary constant a, thus, fiix, p) = a, /^{y, q) = a; § 122.] STANDARD IV. 165 and solve these equations for p and q, thus obtaining p = F^(x, a), q = F,{y, a). Integration of the last two equations gives 2=1 Fi(ps, a)dx ■+■ a quantity independent of x, and '''— \ -^iiyt '*)^2/ + ^ quantity independent of y. These are included in, or are equivalent to z = CFi(x, a) dx +jF2(y, a)dy + b, where 6 is an arbitrary constant. This is the complete integral, since it contains two arbitrary constants ; the general integral and the singular integral, if existing, are to be found as before. Ex. 1. Solve q —p + x — y = 0. Separating q and y from p and x, q-y=p-x. Write q~y=p — x = a; hence p = z + a and q = y + a ; and therefore the complete Integral is 2z=(x + a)2 + (?/ + a)2 + 6. There is no singular integral; the general integral is given by the elimination of a between 22=(x + a)2+ (y + ay + f(a) and = 2(a; + a)+2(y + «)+/'(«), / being an arbitrary function. Ex. 2. Solve _p2 22 = ^~^. Hence «p2 _ x = 3g2 _ y. Put dZ for zhz. Ex. 3. Solve 5 = 2 yp^. Ex.4. Solve Vp + Vq = 2x. Ex. 5. Solve p^ + q'' = x + y. Ex. 6. Solve 22(p2 + g2)=x2+ 2/2. 166 DIFFERENTIAL EQUATIONS. [Ch. XII. 123. General method of solution.* It will be remembered that, in order to solve some of the ordinary differential equar tions of the first order in Arts. 24-29, another differential relation was deduced; and by means of the two differential relations, that were thus at command, the derivative was eliminated and a solution obtained. The general method of solving partial equations of the first order will be found to present some points of analogy to the method employed in the articles referred to. Take the partial differential equation F(x,y,z,p,q) = 0. (1) Since z depends upon x and y, it follows that dz =pdx + q dy. (2) Now if another relation can be found between x, y, z, p, q, such as fix,y,z,p,q)=0, (3) then p and q can be eliminated; for the values of p and q deduced from (1) and (3) can be substituted in (2). The integral of the ordinary differential equation thus formed in- volving x, y, z, will satisfy the given equation (1); for the values of p and q that will be derived from it are the same as the values of p and q in (1). A method of finding the needed relation (3) must now be devised. Assume (3) for the unknown relation between x, y, z, p, q, which, in connection with (1), will determine values of p and q that will render (2) integrable. On differentiating (1) and (3) with respect to x and y, the following equations appear : * This method, commonly known as Charpit's method, in which the non-linear partial equation is connected with a system of linear ordinary equations, is due partly to Lagrange, but was perfected by Charpit. It was first fully set forth in a memoir presented by Charpit to the Paris Academy of Sciences, June 30, 1784. The author died young, and the memoir vas never published. § 123.] GENERAL METHOD OF SOLUTION. 167 dF dF dF dp dF dq ^ 1 p -\ — -\ - = '■'> dx dz dp dx dq dx dx dz dp dx dq dx ' dF dF ,^3p aj'a^_ dy dz dp dy dq dy ' dy dz dp dy dq dy The elimination of -J- between the first pair of these equa- tions gives dFdf dFdf\ f^V_^^\,^f^^_^¥ dx dp dp dx) ^\dz dp 9p dzj dx\dq dp dp dq and the elimination of — between the second pair gives dy dFdf_dFdf\ f^^_^V\,^{^V_^¥\_Q dy dq dq dy) ^\dz dq dq dz ) dy\dp dq dq dp) On adding the first members of these two equations, the last bracketed terms cancel each other, since dq dh 9p_ dx dxdy dy hence, adding and re-arranging, d_F dF\df fdF[ dFl\df f_ dl^_ dF\df dx ^ dz)dp \dy'^^ dzjdq \ ^ dp ^ dq)dz V dp)dx V ^qj^y This is a linear equation of the first order, which the auxil- iary function / of equation (3) must satisfy. This form has been considered in Art. 117, and its integrals are the integrals of 168 DIFFERENTIAL EQUATIONS. [Ch. XII. dp dq dz dx dF dF dF dF~ dx dz dy dz dF dF~ ^dp Uq -dF dp dy df dF (5) dq Any of the integrals of (5) satisfy (4) ; if such an integral involve p or q, it can be taken for the required second relation (3). Of course, the simpler the integral involving p or q, or both p and q that is derived from (5), the easier will be the subsequent labor in finding the solution of (1). This method is applicable to all partial differential equations of the first order ; but it is often better to enquire whether the equation to be solved is reducible to one of the standard forms discussed in Arts. 119-122. The reduction and the subsequent integration by one of the special methods is generally, but not always, less laborious than the integration by the general method. By applying the general method to the linear equa- tion and the standard forms, the integrals obtained in the pre- ceding sections are easily obtained.* Ex. 1. Solve (1) X?' + l) + (6-2)3=0. Here equations (5) Art. 123 reduce to ,a\ dp dq dz dx -'■■ \^) — — ^- — = ■ pq q^ Spq^ +p+(^b- z)q q'^ + 1 -s + b + 2pq dz The third fraction, by virtue of the given equation, reduces to 2pq' From the first two fractions, there comes, on integration, q = ap, where a is an arbitrary constant. This and the original equation determine the values of p and q ; namely, , = ^ct(^-b)-l ^ q^^a(z-b)-l. * See Forsyth, Differential Equations, Arts. 203-207 ; Johnson, Differ- ential Equations, Arts. 288-293. §124.] EQUATIONS OF THE SECOND ORDER. 169 Substitution of these values in dz =pdx+ qdy gives dz = i—-\-dy\ y/a{z- 6)-l, vfliere the variables are separable ; this on integration gives 2^/a{z — 6)— 1 = X + ay + b. There is no singular solution ; the general solution is obtained in the usual way. This equation comes under Standard III. , and the ratios chosen from (2) give the relation q = ap, which is used in the special method. Had there been chosen the equation formed by another pair of ratios from (2), say from dq _ dx q-i "52+1' another complete integral would have been obtained ; namely, Ex. 2. Solve z =pqhy the general method. Ex. 3. Solve (p2 + qi-^y-q^^ Ex. 4. Solve the linear equation and the standard forms by the general m 3thod. Partial Differential Equations op the Second AND Higher Orders. 124. Partial equations of the second order. la this and the following articles,* a few of the simplest forms of partial dif- ferential equations of the second order will be briefly consid- ered ; hardly more will be done, however, than to indicate the methods of obtaining their solutions. Some of these equations are of the highest importance in physical investigations. In what follows, z being the dependent variable, and x and y the independent, r, s, t will denote the second derivatives : * In connection with these articles read the introductory chapter of W. E. Byerly, Fourier's Series and Spherical Harmonics. 170 DIFFERENTIAL EQUATIONS. [Ch. XII. dx'' dx dy dy^ There will be discussed linear equations only ; that is, equar tions of the first degree in r, s, t, which are thus of the form Br + Ss+Tt= r, where R, S, T, V are functions of x, y, z, p, q. The complete solutions of these equations will contain two arbitrary funC' tions.* In Art. 126 will be given some examples of equations that are readily integrable, the special method of solution necessary being easily seen ; and in Art. 126 will be given a general method of solution. 125. Examples readily solvable. It is to be remembered that X and y, being independent, are constant with regard to each other in integration and differentiation. Ex. 1. Solve -^ = ^+a. dxdy y Writing it ^ = ?+a, (ly y integration with regard to y gives p = x\ogy + ay + (pi(x), the constant witli regard to y being possibly a function of x. Integrating tlie last equation with regard to x gives g = j {x log y + ay + i{x)]dx, = '^\ogy + axy + (x) +>/'(«/), the constant with regard to x being possibly a function of y. Ex. 2. Solve -^:f-^/(x)= F(y). dxdy dx-'^ ' ^■" Rewrite it, ' ^ + ^/(a;) = F{y) . * See Art. 109, Ex. 5. § 125, 126.] Iir + SS + Tt= r. 171 This equation is linear in p, and x is constant with regard to y ; hence integration gives p = e-vA»^)| \ey/(='^F(y)dy + 0(a;)l ; and integration of this with regard to x gives a = J je-i'/wr J e»«')J'(2/)d!/ + (^(a:)! |dx+ ^^C?/). Ex. 3. ar = xy. Ex. 4. xr = (n— l)p. 126. General method of solving Br + Ss + Tt= V. On writ- ing the total differentials of p and q, dp = rdx + s dy, dq = sdx+ t dy, the elimination of r and t by means of these from the given equation, Br + Ss+Tt= V, (1) gives (Rdpdy+Tdqdx-Vdxdy) — s{Rdy^~Sdxdy+Tdaf)=0. If any relation between x, y, z, p, q will make each of the bracketed expressions vanish, this relation will satisfy (1). From Rdy''-Sdxdy+Tdx' = 0\ Rdpdy + Tdq dx — Vdx dy = Oj and dz= pdx + q dy, (2) it may be possible to derive either one or two relations between X, y, z, p, q called intermediary integrals, and therefrom to deduce the general solution of (1). For an investigation of the conditions under which this equation admits an interme- diary integral, and for the deduction of the way of finding the * These are called Monge's equations, after Gaspard Monge (1746-1818), the inventor of descriptive geometry, who tried to integrate equations of the form Ilr+Ss+ Tt = 0, in 1784, and succeeded in some simple cases. The method of this article is also called by his name. 172 DIFFERENTIAL EQUATIONS. [Ch. XII. general integral see Forsyth, Differential Equations, Arts. 228- 239. The statement of the method of solution derived from this investigation is contained in the following rule ; Form first the equation Mdy^ — Sdxdy + Tda? = 0, (3) and resolve it, supposing the first member not a complete square, into the two equations dy — mjdx = 0, dy — m^dx = 0. (4) From the first of these, and from the equation Bdpdy + Tdqdx— Vdxdy, (5) combined if necessary with dz=pdx + qdy, obtain two inte- grals Ml = a, Vi = b; then where fi is an arbitrary function, is an intermediary integral. From the second of the equations (4), in the same way, obtain another pair of integrals, mj = a, v.2=b; then is another intermediary integral, /j being arbitrary To deduce the final integral, either of these intermediary integrals may be integrated ; and this must be done when mi = wij. When mi and m^ are unequal, the two intermediate integrals are solved for p and q, and their values substituted in dz =2ydx + q dy, which, when integrated, gives the complete integral. Ex. 1. Solve r — aH =: 0. (This equation is solved by another method in Art. 128.) Here the subsidiary equations (4) and (5) are (1) dy + a dx = 0, dy — a dx = 0, (2) dp dy — ahlx dq = 0. Hence y + ax = ci, y — ax — d. § 127.] GENERAL LINEAR PARTIAL EQUATION. 173 Combining the first of equations (1) with (2), dp + adq = 0, whence p + ag = c'l = Fi,{y + ax); combining the second of (1) with (2), dp — adq = 0, whence j) — aq = c'a = F^^y — ax). From the last two integrals P = K^i(y + «»;) + F^iy - ax)}, and 2 = r- l^iiv + «*) - ^^Cv - «*)]■ 2 a Substitution of these values of p and q in dz =pdx + q dy, gives, on re- arranging terms, dz = — [Fi(y + ax){dy + adx)— Fi^y — ax)(dy — adx)}, 2 a which is exact. Integration gives z = '(y + mx), D"z = m"<^'"' {y + mx), Z>'"2 = <^("' (y + mx), and, in general, that iny'z = m''^<''+'' {y + mx). Therefore, the substitution of (^ (y + moo) for z in the first mem- ber of (1) gives {A^m" + A^m"-'^ -\ 1- A^ <^'"' {y + mx). This is zero, and consequently, 4>{y + mx) is a part of the comple- mentary function if m is a root of § 128, 129.] THE HOMOGENEOUS EQUATION. 175 Am" + Aim'-' + • , • + A= 0, (2) which may be called the auxiliary equation. Suppose that the n roots of (2) are mj, m^, •■•, m„, then the complementary function of (1) is 2 = My + "*i*) + 'l>2(y + 1^^) -I \-4>niy + «i»«), where the functions <^ are arbitrary. The factors of the co- efficient of z in (1) corresponding to these roots are D — mj)', D — rriiD', ■■•, D ~m„D'; and these are easily shown to be commutative. (Compare Arts. 50, 54.) Since e-^c^C?/) = (1 + mxD' + '^D" + -){y), = ^(y) + mxcl>'(y) + 'ffr(y) + -, ==(y+mx), the part of the C.F. corresponding to a root m of (2) may be written e'^"'<^{y). Ex. 1. J^-a2^=0. (See Ex. 1, Art. 126.) Here (2) is m? — cl^^ 0, whence m has the values -fa, — a. Hence the solution is a = 0(2/ + ax) + ^(2^ — ax). Ex.2. Eind the C.F. of ^-|-3-^+2^ = a; + «. Ex. 3. Find the C.E. of ^ — ^ - 6 ^ = xm. 5x2 Qxdy dy^ 129. Solution when the auxiliary equation has repeated or imaginary roots. As in the case of equations between two variables (see Arts. 51, 52), further investigation is required when the roots of (2) Art. 128 are multiple or imaginary. The equation corresponding to two repeated roots m is {D - mD^D - mD') z = 0. On putting v for (D — mD') z, this becomes (D — mD') v = 0,oi 176 DIFFERENTIAL EQUATIONS. tCn. XII. which the solution is v = {y + mx). The Lagrangean equations of this linear equation of the first order are ^^^_dy^ dz m ^ (2/ + mx) Tlie integrals of these equations axe y + mx = a, z = X(t> (a) + & ; and hence, z = x^(y + mx) +il/(y + mx). By proceeding in this way it can be shown that when a root m is repeated r times, the corresponding part of the comple- mentary function is x'-'^2(i/+mx) -\ \-X(l>,_i(y+mx) + 4>Xy+'mx). When the roots of (2) Art. 128 are imaginary, the corre- sponding part of the solution can be made to take a real form.* Ex. 3!£_3-3!^ + 3-5^-3!£ = o. 130. The particular integral. Equation (1) Art. 128 being expressed by F(D, D')z = <^ (aj, y), the particular integral will be denoted by j, J"^ .^ (x, y), ^^^^ V being defined as that function which gives V when it is operated upon by F{D, D'). (Compare Art. 67.) By Art. 128, ^ (x, y) will now be indicated. For this purpose, D — mD' * See Johnson, Differential Equations, Art. 319 ; Merriman and Wood- ward, Higher Mathematics, Chap. "VII., Art, 25. 130.] THE PARTICULAR INTEGRAL. 177 it is first necessary to evaluate {D — mD') (a;, y) = D {x, y — mx). Direct differentiation shows that De-""^.^ {x, y) = e-™^«(Z) _ mD')^ {x, y). From equating the second members of the last two equa- tions, and operating upon these members with e""-°', it follows that (D - mD')4> (x, y) = e"«-°Z)^ {x, y — mx). That a similar formula -^-i-^ ^ (x, y) = e-^i <^ (x, y - mx) (2) holds true for the inverse operator is easily verified. For, the application at D — mD' to both sides of (2) gives d> (x, y) = (D — mD')e-^'''~^{x, y — mx), = {D — mD')e""'>'\l/ (x, y), on putting \p(x, y) for —(x, y — mx); and, therefore, by Art. 128, -^ <^ (x, y)z= (D — mD')\l/ (x, y + mx). But the second member of the last equation is also the result that would be obtained by putting y + mx for y in Dtp (x, y) after the differentiation had been performed ; and this would be (j>(x, y) from the definition of xj/ given above. Hence -(x, y) can be evaluated by the following rule, which D-mD is the verbal expression of (2) : form the function <^{x,y — mx), integrate this with respect to x, and in the integral obtained, change y into y -f mx. 178 DlFPMBENrlAL EQUATIONS. [Ch. XII. The value of the second member of (1) is obtained by applying the operations indicated by the factors, in succession, beginning at the right. Methods shorter than this general method can be employed in certain cases, which are referred to and exemplified in Art. 132. Ex. 2 also shows such a case. Ex. 1. Find the particular integral of Ex. 2, Art. 128. The particular integral "d^ + 3DD' + 2 2>'^ "^^ "•" ^^ = Z> + 2 Z)' ■ dTd^ ^"^ "^ ^^ = e-""'— (2 x+y) = (x''+x ■ y-x) = — xy D ^ ' 32 ^'^ ' 2 3 Ex. 2. Evaluate (.ax + hy). In this case a short method F{D, D') ^ ^ can be used in finding the integral. Since F{D, D')= D^fI—], and —0(aa; + 6;/) = -, and consequently Fi^\'P{ax + hy) = Fi-\ it follows that When^isarootofF(^] = 0, then J^(^] = f^- *)^(^], and the integral is • — ^-— — — (" f... ^rj,{ax + 6)(dx)»-i ; the latter expression can be evaluated by the general rule. Ex. 3. Find the particular integral of ^' - a^ ^ = x^. Ex. 4. Find the particular integral of Ex. 3, Art. 128. §131.] THE NON-HOMOGENEOUS EQUATION. 179 131. The non-homogeneous equation with constant coefficients: tie complementary function. In order to find the complement- ary function of (3) Art. 127, that is, the solution of F{D, D')z = 0, (1) first assume z = ce'"'^"'. (This procedure is like that of Art. 50.) The substitution of this value of z in F{D,D')z gives cF(h,.lc)e'"+'^. This is zero if F(h,k) = 0; (2) and then z = ce'"^'^ is a part of the complementary function. The solution of (2) for k will give values /i(/i), /2(/i), •••,/,(/i), if D' is of degree r in (1). The part of the solution of (1) corre- sponding to k =fi(Ji) is 2cie'"+A('''!', 2 indicating the infinite series obtained by giving c and h all possible arbitrary values ; hence the general solution corresponding to all the values of k is This solution can be put in a simpler form when f(li) is linear in h, that is, when k = ah + b. In particular this is true of the homogeneous equation, which is, of course, a special case of (1). Exs. 2, 3 illustrate these remarks. Equally well may (2) be solved for h in terms of k, and another form of the solution will be obtained, as in Exs. 1, 2. Ex.l. 3!5_a%^o. Here (2) is h^ — Ifi = 0, whence h = J^, and thus the solution is z = 'Sce'"+^*'i, where c and h are arbitrary. Particular integrals are obtained by giving h particular values ; for example, the values 1, 5, | for h give the particular solutions z = ««+!', z = e^'^+^^y, z = 6^"^+?^". If equation (2) be solved for A, the particular integral is 2ce*!'+". Ex.2. 2^'^-5£52_a% , 6^+3& = bx^ dx dy dy^ dx dy Here (2) is IK^ -hlc-h^ + Qh + ?, k = 0, where the values of k are -ihjh + Z; hence 180 DIFFERENTIAL EQUATIOIS S. [Ch. XII. Since each of these series consists of terms liaviug arbitrary coefBcieiits and exponents, it can be represented by an arbitrary function. Conse- quently the solution can be represented by z = 0(x — 2 y) + e^y'pix + y). The equation above might have been solved for h, the values bein" -^,k- 3. Hence, z = Scie'v~i) + se'(i+s)-3j:=0/'j^ _2\ -|- e-^\jj{x + y) is another way in which the solution may be written. Ex. 3. Solve Ex. 1, Art. 128 by this method. Here the values of h are ak, — ak, and hence z = Scie'(!'+'"^) + Scae'C-""^) = ^(y + ax) + ^{y - ax). Ex. 4. Find the complementary function of dx^ 32/2 dx dy Ex. 5. Find the complementary function of fi-T^ + ^-^ = <^os(x + 2y)+ey. dx' dx dy dy Ex. 6. Find the complementary function of 5^_5!£+55+3 3£_20 = c-»-a:22,. dx" dy" dx dy 132. The particular integral. The particular integral can be obtained in certain cases by methods analogous to those shown in Arts. 60-64. It is easily shown, by the method adopted in Arts. 60-62, that ^ e'"+'" = — ^ — 6''"^+'"': that F{D, D') F{a, h) — sin (ax+hif) = — sin (ax+hy), F{D\ DD', D'f ^ ■" F{-a\ -ab, -V) ^ ^" and similarly for the cosine ; and that -— afy' can be •^ F{D, D') evaluated by operating upon x'y' with \_F{D, D'yy^ expanded in ascending powers of D and D'.* * For a full discussion, see Johnson, Differential Equations., Arts. 328- 334. §132.] . EXAMPLES. 181 dx^ dxdy dy^ dx dy The complementary functiun, found by Art. 131, is _D'_2Z)'2 + 2Z> + 2i»' -10 1 i)2 _ j)j)i _ 2 i)'2 + 2 D + 2 i> sin(2a; + j/) ^ sin(23i + y) = l ^ ^' sin(2u: + y) 2(i) + D') "' 2 i>2 - i)'2 _ cos(2 x + y) 6 1 1 1 BH - DD' - 2 D'-' + 2 B + -2 D' ^^ B + S' ' D ~ 2 D' + 2 ' ''^ _1. 1 /^ i)-2i)' (■D-2i?02\ 2 Z) + 2>'i 2 ^ 4 j^ = ^(6a;?/ -6y~ 2x^ + 9x- 12). Therefore, the general solution is « = 0(2/ - K) + e-^, z = r cos Q, (1) becomes :j: a^ 1 5^ 2 aw cote 5;y ^__ a^ br" r" ee' r dr r" 60 r' sip,^ 6 d^' which may be wi'itten I'^Xdry drj sined6\ 86 J sin'O 84,' j ' ^^ and if ;a = cos 9, it will take the form dHvr) a (.^ ^dv) 1 d'v _Q ,^. 8,^ ^8^]^ '^^8^ri-l^'S' ^^ The subject of Spherical Harmojiics is in part concerned with * B. 0. Peirce, Neiotonian Potential Function, Art. 28 ; Thomson and Tait, Natural Philosophy, Art. 491. t W. T. A. Emtage, Mathematical Theory of Electricity and Magnet- ism, p. 14. t Todhunter, Differential Calculus, Art. 207 ; Williamson, Differential Calculus, Art. 323 ; Edwards, Differential Calculus, Art. 532. The equa- tion as given by Laplace was in the form (2). 184 DIFFEBENTIAL EQUATIONS. [Ch. XII. the development of functions that will satisfy this equation.* A homogeneous rational integral algebraic function of x, y, % of the )^th degree, that is, a function of the form f'f{B, <^) in spherical co-ordinates, which is a value of v satisfying (1), is called a solid spherical harmonic of the nth degree ; and f{0, ) is called a surface spherical harmonic of the nth degree. Spher- ical harmonics are also known as Laplace's coefficients.! If V be independent of <^, (3) reduces to On putting v = fP, where P is a function of $ only, and changing the independent variable 6 by means of the relation ju, = cos 6, (6) becomes ^^(l-,^')^ + n(n + l)P=0, (6) which is Legendre's equation. Art. 83. A function that satis- fies (6) or (5) is called a surface zonal harmonic. A particular class of zonal harmonics is also known as Legendrean coeffi- cients.!: For a treatment of spherical harmonics, see Byerly, Fourier's Series and Spherical Harmonics, Chap. VI., pp. 195- 218; and of zonal harmonics, see the same work, Chap. V., pp. 144-194. In special cases (1) and its solution assume simple forms; two of these will now be shown. * See Williamson, Differential Calculus, Chap. XXIII., Arts. 332- 337 ; Edwards, Differential Calculus, Art. 189 ; Lamb, Hydrodynamics, Ed. 1895, Arts. 82-85 ; Byerly, Fourier's Series and Spherical Har- monics. t So called after Laplace, who employed them in determining F in a paper bearing the date 1782. t After Legendre, who first introduced them in a paper published in 1785. Legendre's work in this subject, however, was done before that of Laplace (Byerly, Fourier''s Series and Spherical Harmonics, Chap. IX., p. 267). See Ex. 5, Art. 82. §135.] SPECIAL CASES. 185 135. Special cases. In the first instance given in Art. 134, suppose that the attracting mass is a sphere composed of con- centric shells, each of uniform density. Here v obviously depends only upon the distance of the point P from the centre of the sphere, and hence (2) Art. 134 reduces to dr' r dr ' which on integration gives v = A + -- (2) r Equation (1), in which v depends upon r alone, can be obtained directly from (1) Art. 134 by means of the relation r' = x^ + y'' + z\ For, dv _dv dr _x dv dx dr dx r dr d^v _1 dv_oi^dv,3^ ^v _ da? r dr r^ dr r' dr" ' and on finding similar values for -— > v^j ^^^ adding, there results (1). ^y ^^ For the discussion and integration of (1) from the point of view of mechanics, see Thomson and Tait, Natural Philosophy, Vol. I, Part II., p. 35. If the point P in the second instance of Art. 134 be outside of a uniformly electrified sphere and at a distance r from the centre, obviously — = and — = ; and equations (1) d + (2/2-za;)5 = 22_xj/. 15. j)2 + 52 = npg. in (y — z)P , (2 — x)o X — w ,„ /T— =— 5 . 11. cos(x + y}p + sin(x + y)q = z. 17. Vp + V? = 1. 12. p2 _ 22(-i _ p^) . 18. g = xp + p2. 13. 9=(« + px)2. 19. p(l + ?)=ga. 20. Find three complete integrals of pg =px + qy. 21. (x2 + 1/2) (pi + (f) =1. 22. pg = x"'!/"2'. 23. (X + 2/)(jB + g)2 + (a; - j/) (p - qy = 1. 24. (i/-x)(gy-px) = (i5-g)2. 25. (p + 9)(px + ??/)= 1. 26. X)' + p = 9 x2i/2. 27. s + ?^ = ^. 28. q''r -2pqs +pH=^ 29. g(l + 9)r - (p + 2 + 2 V1> -V vO- + P)t = 0. 30. yr ~{n -\)yp + a. ZZ. p + r = xy. 32. xr-p = x2/. 35. r + (a + b)s + aht = zy. 36. (6 + cg)2r-2(6 + C3)(a + cp)s + (a + cp)2f = 0. S'!'- s + j-^j-j p = aj/3. 38. ?• + ^ s = 15 xy^. MISCELLANEOUS NOTES. NOTE A. A system of ordinary differential equations, of which a part or all is of an order higher than the first, can be reduced to a system of equations of the first order. Take the single differential equation of order n Vdx"' dx"-^' ' dx ' } (1) and put 1^ = 2,1, pL y,,...,iL!M = y„_, ax dx^ dx" ' Then (1) can be replaced by the following system of n equations of the first order, dx dx "" dy„ dx ■ Vn-l, fl (iy«-\ ^XOx"' 2/«-ll2/n-2, •••,2/i, ,2/,xj=0. Again, suppose that there are two simultaneous equations, dz d%^ h[x,y,^, ^ h On putting fM = 2,,, ^^ dx dx^ dx (Jx'^' dx'' ^' dx dx:''} (x V ^ ^ -^ « — —\ = \ ' dx' dx^' C&3' ' cfe' dxy dz dx 189 = zi, these two equations can be 190 DIFFEMENTIAL EQUATIONS. replaced by the following equivalent system of equations of the first order : ay dx = yu dyi dx = y-i, dz dx = 2l, /.(x,y, 2/1, J/2, J, ^, ^i,g) = 0. It is evident that any system of ordinary differential equations can be reduced in this manner to another equivalent system, where there will appear only derivatives of the first order. NOTE B. [This Note is supplementary to Art. 1.] The Existence Theorem. Following is a proof of the existence of an integral of an equation of the first order.* Suppose that the differential equation 0(2/', y, x)= 0, where y' stands for |. is put in the form ,, =^(^, y), " (1) which is always possible. This proof is limited to the case where /(a;, y) is a function which can be represented by a power-series t Oo + aix + a^y + asx^ + aixy + a^y^ + 1- atx'^ H — , in which the a's are all known, since f(x, y) is known, and which con- verges for lal^r, \y\^t, say. (The symbol |x| denotes the numerical value of X.) * This proof is taken from notes of a course on differential equations given by Professor David Hilbert at .Gottingen. t This is by far the most important case, since in the higher mathe- matics such functions are almost exclusively dealt with, and in applied mathematics they are universally used for approximations. MISCELLANEOUS NOTES. 191 It is to be shown tl]at there is a convergent series y = ixo + aiX+a2x'^+— (2) which identically satisfies y' =/(«, y)= ao+ aix + aiy + a^x^ + atxy + a^y'^ + ... + aiX'HI^ + ... ; (3) and which also satisfies a given initial condition, say, that y = yo when X = Xa* That 2/ = when x = may be taken for the initial condition without any loss of generality. For, on substituting xi + xo for x and j/i + ya for ym (1), It becomes 2/i' = 0(xi, 2/i); (4) and it is evident that for 2/1 = ao' + ai'x + 0.2' x'^ + ... to identically satisfy (4) and the initial condition that 2/1 = when xi = 0, is the same thing as for (2) to satisfy (3) and the initial condition that y = yii when x = xo. Hence the initial condition may he taken in this form at the beginning ; and for this it is both necessary and sufficient that tto in (2) be zero. It will now be shown (a) that there is one and only one series, 2/=OiX+ 02X2+ ..., (5) which satisfies (3) identically; and (J) that within certain limits for x this series is convergent. On transforming the series in (3), which has been supposed conver- gent for lx|^)s |!/|^{, by putting x = j-Xi, y = tyi, equation (3) takes the form y' =f{rxu tyi)=ao' + ai'xi + a2'2/i + as'xy' + ai'xiyi + .... The second member of this equation is a convergent series, and con- verges when xi = j/i =1 ; and, therefore, oo' + ai' + a^' + ... converges. This shows that the absolute value of each a' is not larger than a certain finite quantity A, say. The substitution just made for x and y does not make any essential change in the problem, and hence it might have been assumed at first that the «'s of (3) were each not greater than A. In what follows the a's are accordingly regarded as not greater than A. * If an initial condition be not made, then an infinite number of series can be found which will satisfy (3). 192 DIFFERENTIAL EQUATIONS. If (5) satisfies (3), the value of y and y' derived from (5), when sub- stitated in (3), must make the latter an identity; and, therefore, ai + 2 a^x + 3 aaa:^ + • ■ ■ = ao + aix + a2(aix + 02x2 + •■■) + asx^ + aix(aix + ajx^ + ...) + asCaiX + 02X2 + ■■■y+ ... is an identical equation. Hence ai = 00 ; 2 02 = ai + a^au whence 02 = °' + "g"" ; Sas = as + (2202 + aiai, whence 03 = as + ^(ai + 0200) + a4ao ; and similarly for 04, 05, .... It is evident that all the a's can be determined as rational integral functions of the a's ; and it is also to he noticed that all the numerical coefficients in the expressions for the a's are positive ; and, therefore, the a's will not be diminished if each of the a's is replaced by A. From the method of derivation it is evident that (5) with the a's de- termined as above identically satisfies (3). It has still to be determined ■whether this series is convergent. On replacing each of the a's in (3) by A, a quantity not less than any one of the a'S, there results y' = A{l + x+y + x^ +xy + y^ + x^ + x^y+ ■■■). (6) The integral of this equation is found by replacing each of the o's that occur in tlie expressions for the a's of (5) by A. None of these latter coefficients are diminished by changing each of the o's to A, as pointed out above ; hence, if the integral of (6) is convergent, the integral of (3) is also. Now solve (6) directly. On factoring the second member, the equation becomes y' = A(l + x + x^ + -)(i + y + y^+-), 1 1 = A- 1 — X 1 — !/ dx Therefore, (1 — y)dy = A 1-x' whence, on integration, y — ^y^ = — A log(l — x) + ci. Therefore, y = l±[2A log(l - x) + c + 1]1 Here c must be determined, so that the initial condition be. satisfied, namely, that y = when x = ; therefore = 1 ± Vc + 1. Hence the square root must have the minus sign, and c must be zero. MISCELLANEOUS NOTES. 193 Therefore, j,= l-[l+2^1og(l-x)]* = l-[l-2^(x+^ + |+...)]l (7) The series a; + — + — H converges for |a:|< 1 ; hence the square root ofl-2^(x + — + — H J converges for |x|<{y)dy dx dy = d [ F+ <()(!/)], an exact differential. NOTE P. [This Note is supplementary to Art. 49.] On the criterion that n integrals yi, yi, ■■■,yn of the linear differential equation be linearly independent. * I am indebted for this proof to Professor McMahon, of Cornell. 198 DIFFERENTIAL EQUATIONS. Before proceeding to establish the criterion, it may be remarked that if there be a linear relation "1^1 + 022/2+ ■•■=0, (2) where ai, aj, ■•■ are constants, existing between all or any of the integrals Vh 2/2, •■•, Vn, then the integral y = cij/i + 022/2 + ••■ + c"j/„, in virtue of (2), may be written y = lc2 - Cl—]y2 + Us - ci'^\ys + ■■■ +lcn- ci^\y„. This expression d.oes not really contain more than n — 1 arbitrary con- stants, and therefore is not the general integral. Form the determinant 2/2 ■Vn 2/i' (.1-1) Jn-l) where the elements of each row below the first are the derivatives of the corresponding elements in the row above them. This determinant is known as the functional determinant of j/i, 2/2, •••, 2/n, and will be denoted by iJ. The necessary and sufficient condition for the linear indepen- dence of yi, j/2, •", 2/(1 is that R be not equal to zero. Suppose that this condition holds in the case of re — 1 functions, then it holds for n functions. If there be a relation such as (2) between the functions j/i, y^, ■■■, j/,„ then the elements of one of the columns of R are formed from several ■other columns by adding the same multiples of the corresponding elements of these other columns ; and, consequently, R will be identically equal to zero. Conversely, if R = 0, there will be a linear relation of the form (2) between the functions y, t/i, ••■, !/„. Since R = 0, the determinant must be reducible to a form wherein all the elements of one column are zero ; that is, there must be certain multipliers Xi, X2, ••■, A„, such that ^i2/i + ^22/2H hX„2/„ = Xi2/i' + X22/2' + --- + X„2/„' = Xi2/i<»-" + X22/i<"-" + ••• + Xn2/«<'-« = (3) Differentiation of each of these equations and subtraction of the one next following gives MISCELLANEOUS NOTES. 199 Xj/j,j(n-2) + A2'j,2("-2) + ... + X„'2/„("- If one of the determinants 2/1 2/2 ?/i' 2/2 ^if" -2) ^2( (4) vanishes, say the -Vn' one formed by omitting the rth column, then hy hypothesis there is a relation ClVl + C22/2 H \- Cr-Wr-l + Cr+l2/r+l + 1- C„y„ = 0. But if no one of these determinants vanishes, then it follows from (4) and the first (n — 1) equations in (.3) that ^ = v=... = v Xi X2 x„ Suppose that each of these fractions is equal to p, say. It follows from integration that \i = aieS'"'', K^ = a^eS''''^, ■•■, X„ = a„eJ''''-', ai, a^, ••■, a„ being the constants of integration. On substituting these values in the first of equations (3) and dividing by the common factor eJ"''''^, there appears the relation , , , n ^^ 012/1 + a22/2 H h a„yn = 0, which is thus a consequence of R being equal to zero. Hence, if the criterion holds for re — 1 functioris, it holds also for re. But it can be shown as in Note D that the criterion holds for 2 functions ; hence it holds for 3, hence for 4, and so on for any number. Therefore, the necessary and sufficient condition that y^, j/2, ••■, Vn form a system of linearly independent integrals, or a fundamental system of integrals, as it is sometimes called, is that the determinant E do not vanish identically. NOTE G. The relations between the coefficients of a linear differential equation and its integrals. Let !/i, j/2, •••, 2//. be n linearly independent functions of x. It is required to form the differential equation which has these functions for its inte- grals ; in other words, to form the equation which has 2/ = Ci2/i + C2i/2 H hc„2/„ (1) 200 DIFFERENTIAL EQUATIONS. for its general solution. The difierential equation is fonned by elimi- nating ci, C2, ■••, c„ from tlie given integral by the method shown in Art. 3. By difEerentiatiug n times there is obtained the set of n + 1 equations, y = ClJ/l + C22/2 + — + c„2/„ yi = ayi' + ciJ/i' + ••• + c„yj J/lC) = Ci2/i(») + C2j/2<'' + ■•• + CnVn^"'). Krom this the eliminant of the c's is found to be y yi Vi y' yi' yJ :0, (2) tlie differential equation required. Now suppo.se that the differential equation having the integrals yi, 2/2i ""i yn is in the form |:i+P,|r^+P,^+... + P„(,) = o. (3) On denoting the minors of y, y', ■■■, 2/(») in (2) by T, Ti, •••, T„, respec- tively, (2) on expansion becomes r„f:i^-r„_/— U... +(-!)« r2/ = o. (4) dx" da;»-i Comparison of (3) and (4) shows that Pi=-^, P2=-%=■^■••,P„=(-1)"•|^■ J- n Jn JLn It will be observed that y„ is the determinant B of Note F. *In pg,rtlc- ular, since differentiation will show that r„-i = ^^, Pi = — Ll^; dx T„ dx and hence 7„ = e~J^i<'-". NOTE H. [This Note is supplementary to Art. 102.] On the criterion of integrability of Pdx + Qdy + Bdz = 0. It has been shown in Art. 102 that the necessary condition for the existence of an integral of Pdx + Qdy + Bdz = (1) * This deduction is due to Joseph Liouville (1809-1882), professor at the College de France. MISCELLANEOUS NOTES. 201 is that the coefficients P, Q, M, satisfy the relation pm_dB\+g(dB_dPWM(dP-d^) = 0. (2) \dz dy) ^\dx d^r \dy dxl ^^ It ■will now be proved that this condition is also sufficient, by showing that an integral of (1) can be found when relation (2) holds. Substitution sliows that, if relation (2) holds for the coefficients of (1), a similar relation holds for the coefficients of lj,Pdx + fj.Qdy + ixnde = 0, (3) where /x is any function of x, y, z. If Pdx + Q,dy is not an exact differ- ential with respect to x and y, an integrating factor ]i. can be found for it, and (3) can then be taken as the equation to be considered. Hence there is no loss of generality in regarding Pdx + (^dy as an exact differential. On assuming then that 5^ = 55, by dx and that F = j*(P(ix+ Qdy), it follows that dx ^ dy dP_ d^v^ dz dzdx 32- d^^ dz dzdy Hence, from (2), dVlff'V dE\ dJ^/dR _ i!£\ ^ dxKdzdy dy) dy\dx dzdx) This may be written dL. dx d_ldVi_ 3y\ dz ) dy dy\dz J dx .dV dy dx\ dz d_(dV_ii\^^ dx\ dz b\ = o, = 0. This equation shows that a relation independent of x and y exists between r and ^ - R. dz 202 DIFFERENTIAL EQUATIONS. Therefore, sX. — n can be expressed as a function of z and V alone. dz Suppose that ^F - iJ = (3, T). (4) ds Since Pdx + Qdy + Bdz = ^dx +^dy + ^dz + ( B - ^]dz, dx dy dz \ dz I equation (1) may tie written, on taking account of (4), dV-(z, V)dz = 0. This is an equation in two variables. Its integration will lead to an equation of the form F(V «1 = Hence (2)* is both the necessary and sufficient condition that (1) have an integral. tNOTE I. Modern Theories of Differential Equations. Invariants of Differential Equations. The two modern theories of diiierential equations are : (a) The theory based upon the theory of functions of a complex variable ; (6) The theory based upon Lie's theory of transformation groups. The study of differential equations, until about forty years ago, was restricted to the derivation of rules and methods for obtaining solu- tions of the equation and expressing these solutions in terms of known functions. Even at the beginning of the present century,} however, * Of course this criterion is included in the criterion for the general case of p variables, the deduction and proof of which is to be found in Forsyth, Theory of Differential Equations, Part I., pp. 4-12. (See foot- note, p. 138.) See Serret, Calcul Integral (edition 1886), Arts. 785-786. t Two historical articles that the student would do well to consult are : T. Craig, " Some of the developments in the theory of ordinary differ- ential equations between 1878 and 1893," Bulletin of N. Y. Math. Soc, Vol. 11. (1892-1893), pp. 119-134; D. E. Smith, "History of Modem Mathematics " (Merriman and Woodward, Higher Mathematics, Chap. XI.), Art. 11. Also see E. Cajori, History of Mathematics, pp. 341-347. } " Gauss in 1799 showed that the differential equation meets its limitations very soon, unless complex numbers are introduced." MISCELLANEOUS NOTES. 203 mathematicians saw tliat any marked advance in tbis direction was im- possible witbout the aid of new conceptions and new methods. But it was not until a comparatively recent date, that wider regions were dis- covered and began to be explored. " A new era began with the foundation of what is now called function- theory by Cauchy, Riemann, and Weierstrass. The study and classifica- tion of functions according to their essential properties, as distinguished from the accidents of their analytical forms, soon led to a complete revolution in the theory of diHerential equations. It became evident that the real question raised by a differential equation is not whether a solution, assumed to exist, can be expressed by means of known func- tions, or integrals of known functions, but in the first place whether a given differential equation does really suffice for the definition of a func- tion of the independent variable (or variables), and, if so, what are the characteristic properties of the function thus defined. Few things in the history of mathematics are more remarkable than the developments to which this change of view has given rise." * The leading events in the early history of this new theory are : the publication of the memoir on the properties of functions defined by dif- ferential equations, by Briot and Bouquet in the Journal de VlScole Polytechnique (Cahier 36) in 1856; the paper on the differential equation which satisfies the Gaussian series, by Riemann at Gottingen in 1857 ; and, perhaps, most important of all, the appearance of the memoirs of Fuchs on the theory of linear differential equations with variable coeffi- cients, in CrelWs Journal (Vols. 66, 68) in 1866 and 1868. t The only work in English which employs the function-theory method in discussing differential equations is that of Professor Craig. J A knowledge of the theory of substitutions, as well as of function- theory, is required for reading some of the modern articles on differential equations. * See G. B. Mathews, a review in Nature, Vol. LII. (1895), p. 313. t Albert Briot (1817-1882); Jean Claude Bouquet (1819-1885); Georg Friedrich Bernhard Riemann (1826-1866), the founder of a general theory of functions of a complex variable, and the inventor of the sur- faces, known as " Riemann's surfaces"; Lazarus Fuchs (born 1835), professor at Berlin. X T. Craig, Treatise on Linear Differential Equations (Vol. I., published in 1889). See Note J for the names of other works on the modern theories. 204 DIFFERENTIAL EQUATIONS. Professor Lie * of Leipzig has discovered, and since 1873 has developed, the theory of transformation groups. This theory bears a close analogy to Galois' theory of substitution groups which play so large a part in the treatment of algebraic equations. By means of Lie's theory it can be at once discovered whether or not a differential equation can be solved by quadratures.! An elementary work by Professor J. M. Page on differ- ential equations treated from the standpoint of Lie's theory has been published, t The theory of invariants of linear differential equations is one of the later developments in the study of differential equations. While it plays a very important part in both of the modern theories referred to above, yet, to some extent, it can be studied without a knowledge of these theories. § It has been found tliat differential equations, like algebraic equations, have invariants. An invariant of a linear differential equation is a function of its coefficients and their derivatives, such that, when the dependent variable undergoes any linear transformation, and the in- dependent variable any transformation whatsoever, this function is equal to the same function of the coefficients of the new equation multipled by a certain power of the derivative of th^ new independent variable with respect to the old. The introduction of invariants into the study of differential equations is due to E. Laguerre of Paris. || Those who have made the most im- * Sophus Lie was born in Norway and educated in Christiania. He has been Professor of Geometry at Leipzig since 1886. He has expounded his theory in the following works : Theorie der Transformationgruppen, Vols. I., II., III. (1888-1893) ; Vorlesungen uber continuierliche Gruppen (189.j). See p. 207 for his work on Differential Equations. t For an elementary introduction to Lie's theory of transformation groups, and its application to differential equations, see articles by J. M. Page?: "Transformation Groups," Annals of Mathematics, Vol. \'I1[., No. 4 (1894), pp. 117-133; "Transformation groups applied to ordinary differential equations," Annals of Mathematics, Vol. IX., No. 3 (1895), pp. 69-69. Also see J. M. Brooks, "Lie's Continuous Groups," a review In Bull. Amer. Math. Soc, 2d Series, Vol. I., p. 241. t By The Macmillan Co. § See Craig, Linear Differential Equations, pp. 19-22, 463-471 ; and the memoir of Forsyth referred to below. II In his memoirs: "On linear differential equations of the third order," Oomptes Bendus, Vol. 88 (1879), pp. 116-119 ; " On some invari- ants of linear differential equations," Ibid., pp. 224-227. MISCELLANEOUS NOTES. 205 portant investigations on these invariants are Halplien and Professor Forsyth. Their memoirs* are among the principal sources of infor- mation on the subject. NOTE J. Works on Differential Equations. A brief list of books on differential equations may be interesting and useful to those who intend to continue the study of the subject. For convenience these books are divided into three groups. The first two groups consist of works which have been written from the older point of view ; the third contains works in which any of the modern theories of functions, substitutions, transformations, invariants, etc., are more or less discussed. The first group is made up of the smaller and more ele- mentary works ; the second includes the larger and more advanced. I. OsBOitNE : Examples of Differential Equations with Rules for their Solu- tions, vii + 50 pp. Boston, 1886. Byerly : Key to the Solution of Differential Equations, being pp. 296- 339 of his Integral Calculus, edition of 1889. Boston. Edwards : Elementary Differential Equations, being Ohaps. XIII.-XVII., pp. 211-277 of his Integral Calculus for Beginners. London, 1894. Johnson : Differential Equations, being Chap. VII., pp. 303-373, of Merriman and Woodward, Higher Mathematics. New York, 1896. Stegemann • Integralrechnung (edited by Kiepert), Chaps. XIII.-XV., pp. 407-563, 6th Aufl. Hannover, 1896. 1st Aufl., 1863. Airy : Partial Differential Equations, viii + 58 pp. London, 1866. II. De Morgan : Differential and Integral Calculus, Chap. XI., pp. 183-215 ; Chap. XXI., pp. 681-736. London, 1842. * G. H. Halphen (1844-1889) of the Polytechnic School in Paris. "M^moire sur la reduction des Equations diff^rentielles lineaires aux formes intdgrables," Memoires des Savants Strangers, Vol. 28 (1884), pp. 1-301. Chap. III., pp. 114-176, in this memoir treats of invariants. A. R. Eorsyth, " Invariants, Covariants, and Quotient-Derivatives asso- ciated with linear differential equations," Phil. Trans. Boy. Soc, Vol. 179 (1888), A, pp. 377-489. 206 DIFFERENTIAL EQUATIONS. Price : Infinitesimal Calculus, Vol. II., pp. 51-3-707. London, 1865. Hymees : A Treatise on Differential Equations, viii + 180 pp. 2d edi- tion. London, 1858. Boole : A Treatise on Differential Equations, xv + 496 pp. London, 1859. The same, new edition with supplementary volume, xi + 235 pp., by I. Todhunter, 1865. FoKSYTii : A Treatise on Differential Equations, xvi + 424 pp. London, 1885. Johnson : A Treatise on Ordinary and Partial Differential Equations, xii + 368 pp. New York, 1889. MoiGNo : Calcul Integral, pp. 333-783. Paris, 1844. DuHAMEL : Calcul Infinit&imal, t. II., pp. 122-490. 2d edition. Paris, 1861. Serket : Calcul Integral, pp. 343-676. 1st edition. Paris, 1868. Houel: Calcul Infinit&imal, t. II. (1879), pp. 287-472; t. IIL (1880), pp. 1-237. Paris. Laurent: Traits d'Analyse, t. V., pp. 1-320; t. VI., pp. 1-223. Paris, 1890. BoussiNESQ: Cours d'Analyse Infinitesimal, t. II., 1, pp. 177-229; t. II., 2, pp. 1-7, 229-535. Paris, 1890. Dn Bois-Rkymond : Beitrage zur Interpretation der partiellen Differ- entialgleiohungen mit drei Variabeln. Heft I. Die Tlieorie der Characteristiken, xviii + 255 pp. Leipzig, 1864. RiEMANN : Partielle Differentialgleichungen und deren Anwendung aut physikalische Fragen (edited by Hattendorff, 3d edition, xiv + 325 pp.). Braunschweig, 1882. III. Forsyth : Theory of Differential Equations, Part I. Exact Equations and Pfaff's Problem, xiii + 340 pp. Cambridge, 1890. Demartees : Cours d'Analyse, t. III., pp. 1-134 (in 4° lith.). Equa- tions differentielles et aux differences partielles. Paris, 1896. KoENiGSEERGEH : Lehrbuch der Theorie der Differentialgleichungen mit einer unabhangigen Variabeln, xv -|- 485 pp. Leipzig, 1889. Jordan: Cours d'Analyse, t. VIL, pp. 1-458. 1st edition 1887, 2d edi- tion 1896. Paris. Picard: Traite d'Analyse, t. II. (1893), pp. 291-.347 ; t. III. (1896), xiv + 568 pp. Paris. PjlInleve: Le^;.ons sur la Theorie Analytique des liquations Differ- entielles, 19 -I- 6 -f 589 pp. (Lith.). Paris, 1897. MISCELLANEOUS NOTES. 207 LiE-ScHEPFEKs : VorlesungeH fiber Differentialgleichungen mit bekannten infinitesmalen Transformationen, xiv + 568 pp. Leipzig, 1891. Page : Ordinary Differential Equations, an elementary text-book with an introduction to Lie's Theory of the Group of one Parameter, xviii + 226 pp. London and New York, 1897. GooRSAT ; Lefons sur I'integration des Equations aux d^riv^es partielles du premier ordre, 354 pp. Paris, 1891. GouRSAT : LeQons sur I'integration des Equations aux d^riv^es partielles du second ordre, t. I., viii + 226 pp. , Paris, 1896; t. IL (en cours d'impression), 1897. PocKELS : Ueber die partielle Differentialgleichung Am + k^u = und deren auftreten in der mathematischen Physik, xii + 339 pp. Leip- zig, 1891. Mansion-Masbr : Theorie der partiellen Differentialgleichungen erster Ordnung, xxi + 489 pp. Berlin, 1892. PoiNCARE : Sur les Equations de la physique math^matique, 100 pp. Paris, 1894. Painleve : Lemons sur I'integration des Equations difl^rentielles de la mfcanique et applications, 4to. (Lith.) 295 pp. Paris, 1895. Craig: Treatise on Linear Differential Equations, Vol. I., ix -)- 516 pp. New York, 1889. * Hepftek : Einleitung in die Theorie der linearen Differentialgleichungen mit einer unabhangigen Variabeln, xiv + 258 pp. Leipzig, 1894. Klein: Vorlesungen tiber die hypergeometrische Function, 571 pp. (Lith.) Gbttiugen, 1894. Klein : Vorlesungen tiber lineare Differentialgleichungen der zweiten Ordnung, 524 pp. (Lith.) Gottingen, 1894. t ScHLEsiNGKF : Haudbuch der Theorie der linearen Differentialgleichun- gen, Bd. I., XX + 486 pp. Leipzig, 1895. Bd. II., Th. 1, xviii -f 532 pp. 1897. * See review by M. B6oher, Bull. Amer. Math. Soc, 2d series. Vol. IIL, p. 86. t See review of Vol. I. by G. B. Mathews, Nature, Vol. LIL, p. 313 ; and review by Bocher, Bull. Amer. Math. Soc, 2d series. Vol. III., p. 146. 208 DIFFERENTIAL EQUATIONS. NOTE K. [This Note is supplementary to Art. 53.] The Symbol D. Let Z) be a symbol which represents differentiation, with respect to X say, on the function immediately following it. In other words, let Du = — (tt), or Du = —- (1) It is evident from definition (1) and the results (2), (3), that the result of the operation symbolized by D taken n times in succession will be d''u dx" Also, let the operations which consist of the operation D repeated two, three, •••, n times in succession be denoted by D^, D', •■•, V". It should be noted that, according to this definition. D"u represents — - and not /d«\" ' ''*" I — I . From this definition of D" it follows that the operational symbol \dx/ D is subject to the fundamental laws of algebra. For, dx'' dx'' dx'' dx' D'.D''u=^l^' "uX ^ d^+'u ^d" /d'u\ J,, j,^ . 'x"! dx"-^' dx^\dvi ' dx'\dx jy ■ D^n - D"+<-u ; D'"(u + ?))=— (u + v) = '^—^ + — = D<»u + D'"v. dx"' dx"' dx" Since D represents an operation, it can only appear with integral exponents. Negative exponents will now be considered. Suppose that Du = v, (4) and let u be indicated by a = D-^v. (5) It is necessary to give a meaning to Z)-', and this meaning must not MISCELLANEOUS NOTES. 209 be inconsistent with tlie definition of D. On operating on each member of (6) with D, Du = D- D-^v, whence by (4), D ■ D-H = v. Therefore JD-i represents such an operation on any function that, if the operation represented by D be subsequently performed, the function is left unaltered. Hence the operation represented by D~^ is equivalent to an integration. It follows that the operation indicated by Z)-" is equivalent to n successive integrations. The proof that the symbol D with negative exponents is subject to the laws of algebra, is similar to that used for D with positive exponents. It has been seen that D ■ D'h) = v. But D-i ■ Dv = v + c, in which c is an arbitrary constant of integration. Therefore, in order that Dm . D-^v = D-" • D'^v, it is necessary to omit the arbitrary constant that arises when the opera- tion indicated by D~^ is performed. NOTE L. [This Note is supplementary to Art. 82.] Integration in series. The law for the exponents will be apparent on substituting k" for y in the first member of the given equation. Suppose that the expression- obtained by this substitution is ./i(m)a;"''+/2(m)a;"'". (3) In general (3) will contain more than two terms ;. in the case of the equations in Art! 66 it contains only one term. Under the supposition just made, the successive differences of the exponents of x in the series sought must evidently be m" — m'. This common difference will be denoted by s. Solution (2) may now be written y = A((e^ + ^ia;»(+' + ••• + ^,_ia;'»+('-i'' + A,x''+'" + ■■•, (4) r=Qo or simply 2/ = 2 ArX?^+". r=0 210 DIFFEBENTIAL EQUATIONS. Substitution of this series for y in (1) will give, in virtue of (3), -4o/i (m) «•»' + Aafi (m) a;"''+» + Aifi{m + s) «'»'+» + Aif2{m + s)x'»'+2' + ■■• + Ar-ifi [m + (»• - 1) s] a?»'+('-i)' + ^-i/a [m + (r - 1) s] a;'»'+" + Arfii^m + rs)x™'+" + ^r/2(»» + »-s) a;"''+ ('+!)• + ••• :0. (6) Since equation (5) must be an identity, the coefficients of each power of X therein must be equal to zero ; hence /i^m)=0 (6) and ArMm + rs) + A^-if^ [m + (r - l)s] = 0. (7) The roots of (6) give the initial exponents of series that will satisfy (1); and equation (7) shows that fx{m + rs) which is the relation between successive coefficients. The difference between the exponents in (3) might have been taken, m' — m" or — s ; in this case, the resulting series would have had their powers in reverse order to those of (4) ; and the initial terms would have been found by solving /2(m)= 0. In determining the initial power of x for an equation of the nth order, that coefficient in (3) which is of the wth degree in m must be put equal to zero, since there must be n independent series in the general solution. If both /i(m) and /aCm) are of the reth degree, two sets of series can be derived, one in ascending powers and the other in descending powers of x. If the expression (3) have another term /3(m)x'»"', the terms of the series can be successively deduced, but the process will be much more tedious. This method can also be employed in the case of non-linear equations, but more than a very few terms can be calculated only with difficulty. The equations previously considered can of course be inte- grated in series ; Ex. 3, Art. 82, illustrates this. ANSWERS TO THE EXAMPLES. L CHAPTER I. (p stands for ^. ) \ dx / Art. 3. 2 2. (pfi-2y^)p^-ipxy-x^ = Q. 3. (1 +i>2)' = J-2^|^\ 4. xyp^ + xm'-y^=0. dx' \dxj dx Page 11. 1. dx _ dy 8. xfl+2^-xy = 0. dx^ dx 9. 5+-, = 0. VI - a;2 VI - y^ 2. i)Vl -x^ = y. 3. da;' dx '«■ -g-(l)'=«' 4. l2p'h/={Sp^-21)x. 11. y^ = 2pxy + xK 8. 6. 7. y=px+p-p^. 8 opS = 27 V. p2(l-a;2)+l =0. 14. ds>;2 + 2y = 2xf. dx CHAPTER II. Art. 8. 2. i/Vl - x2 + a;Vl - j^^ = c. 3. 2/ = c(a + k) (1 - aj;). 4. tan !/ = c(l — e"')'. p 211 212 ANSWERS. Art. 9. 2. xy^ = c^(x + 2y). 3. y^ce'"'. Art. 10. V A^ 1. (y-x + l)\y + x-iy = c. ---^ ^ j_ n ^9 r , 9,-,i ,^ i,N f22/-(5+-v/-2T)x + 2f2+-\/2l)1 v^JT l22/-(5-v'21)x+2(2-\/2l) ^ Art. 13. yZ 3. a'^x — ^ — x!/^ — x^y = c. 4. ax'' + hxy + cy^ + gix + e^ = A. 5. xV -I- 4 x'!/ — 4 XJ/' + 'f — x0i + e^^ + x* = c. Art. 16. 3. 2 a log X + a log 2/ — 2/ = c. 4. xV + my"^ = cx'^. 5. x^ + - = c. Art. 17. 1. - + log?^ = c. 3. 21ogx-log2/= — +c. y X- 1. e«(x'' + 2/'^) = c. Art. 18. 2. x^ — y^ = ex. 4. XJ/ + 2/2+^ = c, 2/^ Art. 19. 2. 5 x~Jt2/H _ 12 x'iiy'ii = c xy 3. x^y^ + x'' = cy. 3. 6Vx!/ — X ^yi = c. Art. 20. 2/=(x + c)e-''. 3. 2^ = tanx-l + ce"""*. 4. j^ = (e"^ + c) (x + 1)". 5. 3(x2 + l)y = 4x3 + c. Art. 21. 3. 7 y~i = cxt - 3 x'. 4. 2/^ = c(l - x2)i - i-^- 5. yS = aj -I- cxVl - x^. A])fSWEItS. 213 Page 29. J a; + y _ ^^^ y + c ' 4. tan a; tan y = *. a ~ o 2. x2 = 2c2/ + c2. 6. log^-?^=c. 2 V ce«y 6. y = x2(l + ce'). 7. 60 2/3(a; + l)2=10a;'i + 24a;6 + 15x4 + c. 8. x^ ~xy + y^ + X — y = c. X 9. y = ce >/i^^ -{ ^ X 10. ca; = e*. 11. a;* - y* + 2 a;V - 2 aV - 2 6V = c. 12. j/(a;2 + l)2 = tan-ix + c. , „ 3 16. logVai^ + ys — TO tan-i- = c. IS. logcy = ^^. * ^ 17. r = ce'»9. 14. y = ox + cxVl — a;2. = ca;. 19. 2/ "+1 = ce*"-') »i" » + 2 sin x + 15. x^ — t/^ — l= ex. 18. x + yey = c. 2 (.+ l)ev = 2. + c. 23^ , + 22,B + f_3,.,. = c. 3 2 21. l = a:2 + l + ce»". »/2 ^ ■ 24. 2/4+2aV+a;*-2a2a;2+2xy=c. i2.?^ = t + c. S85. 1 = 2-2,^ + ce-K n + 2 a; - o . 26. y^ = ax^ + c''. 27. (a; + Va^ + x^)y = a!' log(x + .Va^ + x") + c. 28. log Vx^ + 2/2 + tan-i =^ = c 29. (4 6= + 1) 2,2 _ 2 a(sin a; + 2 6 cos a;) + ce-^. 30. a;2 + y2 _ c!/. 1 32. c(y-6): 31. 3 2/2-2a;2e>^ = ca;2. ' ^ ^ 1 + 6a; 33. 9 log(3 2/ + 2 a; + -2f ) = 14(3 2/ - I a; + c). 34. a;V — 2 a;y log C2/ = 1. 214 ANSWERS. 35. ^x^y = o + cxwr: S'- ^ = ax + c. y 38. «log '^-y + '' +y = c. 36. cj/ = e". 2 x — y ~a CHAPTER III. Art. 22. 3. y = c, X + y = c, xy + x^ + y'^ = c. 5. x^ + 2 y^ = c, x* + y^ = c. 4. 343(1/ + c)3 = 27 aa;'. 6. ?/ = 4a; + c, ?/ = 3x + c. Art. 24. 2. a;+c= -{log (1 + ^2)- J^ (i) — 1) — tan-ip}, with the given relation. 8. log (» — »;)= — h c, with the given relation. p — x 4. 2y = cx^ + -- c Art. 25. 1. y = c-{_p'' + 2p + 2\og(p-l)-\, a; = c-[2p + 21og(i)-l)]. 2. 2/ = c — a log (p — 1), X = c + o log — ^ — • 3. y^ = 2cx+ d^. Art. 26. ■1. X = logjB^ + 6p + c. i./L ■ X 2. y — c= Vx — X- — tan-i -v 3. 2!/ + c = a [pVl +i)2_iog(p + Vl+i)2)], a; = avT+^. 4. a; + c = alog(p + Vl +p2^, 2/ = aVl +^''.. Art. 27. 2 4 Art. 28. 3. y = ca; + sin-i c. 4. e^" = ce^* + c'^. 6. ^^ = g-ga + i ^. c, Page 38. 1. sin-i - = log ex. 2. 2/ = c(a; - 6) + -• ANSWERS. 3. Qx? -y^ + c) (x2 -y^ + ex*) = 0. 5. (y - 6x + c)iy - 3x + c) = 0. 4. xy = c + cH. 6. ac'^ + c(2 x - 6) - j/^ = o. 7. fl + c = f — '^^'^^ ^ ■ , where = tan-i ^, and r^ = x^ + u^. J rVj'2 - 02(,^) x' 8. tan-i - + c = vers-12 a Vx* +«2. X « 1 9. sin"i" + sin-i- = c. X X 10. x2 + 1/2 - 4 ex + 3 c2 = 0. (Put x2 - 3 2/2 = .b2.) 11. x2 + 2,2 + 2 c(x + 2/) + c2 = 0. i±-i/— 14. 2/2 - cx2 + -^^ = 0. 12. y + V2/2 + rax2 _ c^ \ n . c + 1 13. 2/2 — & = (x + c)2. 15. 2/(1 ± COS x) = c. 16. (2/ — sin-1 — c] [2/ — cos"^ — cj ( 2/^ — X2 — c J = 0. 17. (y + c)2 + (x - a)2 = 1. 19. (y - cx^) (2/2 + 3 x2 - c) = 0. 18. y = 2 c-^/x +/(c2). 20. y = c(x - c)2. 21. (xS-32/+c)(e2 +C2/)(X!/+ C2/ + !) = &. 22. ax + e = f p2 — mp + m'^ log (p + m), with the given relation. 23. ee = ce" + c^. 25. (x + c)2 + ()/ - 6)2 = 1. 24. log- I____=c±j/.' 26. 2/ = c»; + — • X + ■\/x2 — 2/2 " 27. 2/2 = cx+|c3. CHAPTER IV. Art. 33. 2. 2/ = ex + c2, x2 + 4 2/ = 0. 4. x2(2/2 - 4 x^) = 0. 3. (2/ + X — c)2 = 4 X2/, X2/ = 0. 5. (x— 2/ + c)3 = a(x+2/)2, x+2/=0. Page 49. 1. 2 2/ = cx2 + " 2/2 = ax2. 2. a'x + CX2/ + c2 = 0, singular solution is x(x2/2 - 4 a') = 0. x = is also a tac-locus. 8. (j/ - cxY = m2 + c2, 2/2 + w2a;2 = m^ 216 ANSWERS. 4. y = cx + -v/62 + aic\ bV + qPy'^ - a%''. 5. y = cx — c'^, x^ = 4Ly. 6. y'^ = iaix — b); l.+ iz'^y = 0, x = is a tac-locus; 27y = ix^; y^ = 4: mx. 7. (j/ + c)2 = a;' ; a; = is a cusp locus ; tliere is no singular solution. 8. (y + c)2 = a;(x - 1) (a; - 2) ; singular solutions are a; = 0, x = 1, I x = 2; x = l ± — — are tac-loci. The curve when c = consists V3 of an oval cutting the axis of x at the origin and at a; = 1, and a curve resembling a parabola in shape, having its vertex at the point for which x = 2. 9. x{x' — c(y + c)2} = ; singular solution is 4 J/S + 27 x' = ; x = is a part of the general solution, and is the cusp locus for one pan of the general solution and the envelope locus for the other part. 10. y = cx + V^^~+¥, bV + aP-y'^ = a^"^. 11. x2 + 2/2-c(x2-2,2)_ 1 + c2 = 0; that is, -^ + -^=1. Singu- 1 + c 1 — c lar solution is x* — 2 xV + !/* — 4x2_4i,2^.4_o. ^^^ jg^ (x + 2/+V^)(x + 2/-V2)(x-2/+V2)(x- 2/--v/2)=0. Tlie general solution is the system of conies touching these four lines. CHAPTER V. Art. 43. 8. y/ny = x + c. 4. 2/ = Vox - x^ + - vers-i — ■ 2 a 6. r = c — K cos ff; when c = k, the cardioid r = k(1 — cos S). 9 6. cr = e". Art. 47. 3. The ellipses 2si!' + y^ = c^. 4. yi - x^ = cl 2 r 6. The confocal and coaxal parabolas r = - 1 — cos 9 25 6. sec5fl + tan59 = ce'. 7. r = ce-*">'9, r = Vx" + i/^, 4> = tan-i-- Art. 48. 2. s = JaJ2 + «o« + so. 3. s = ^jr{2. Page 60. 1. y = ce<"'. 2. xt + j/t = oi 3. Zy'^ = 2K(x^-\-c). ANSWERS. 217 i. s = K siu 0, the intrinsic equation of a cycloid referred to its vertex, the radius of the generating circle being ^ k. 5. The lines }• sin (e + a) = -, and their envelope the circle r = — 6. The parallel lines (rasino — ncoso)* — (mcosa + ?isina)2/ = c. 8. The system of circles passing through the given point and having their oeatres in the given line. 9. x^ + y'' = 2 a' log x + c. 10. x^ - j/2 = c^. 11. r" = c" sin nS; r^ = c^ sin 2 8, a, series of lemniscates having their axis at an angle of 45° to that of the given system. 12. r^ - k'^ = cr cosec 0. 13. r = ^■^-v- 14. Parahola («/ - x)^ - 2 a(^y + x) + a^^ = 0. 15. a;2 + t = a^- 16. x^ + 2/2 = 2 ex. 17. log -^ ((/ + Vi/-^ - x2) = 4 (2^ + V2/2 - x2). x° x^ 18. xV' = c. 19. r» = c sin n9; r = c sin 9 ; »• = c(l — cos &) . 20. )- = c(l -cos9). "{^^i r v^ 4- ^^ 22. re « = c. Vj/ - v^ 23. r = «(1 - «") 24. )- = c(l + cos«). ' 1 — esin(e + c) 27. The conies that have the fixed points for foci. 28. Tlie conies that have the fixed points for foci. 29. The ellipse kV + ah)"- = O^k^. 30. The hyperbola 2xy = a"^. 31. The parabola aV^ = k^{2 ay + k^). 32. The catenary y = a cosh^- (See Johnson, Diff. Eq., Art. 70.) a 33. 4 at/ + c = 2 axV4 a^x^ - 1 - log (2 ax + ^40%'^ - 1). 34. (a) i=:ce ''' +-e ^' J e''' fCt)dt. L (b) i = ce ^ . If i = /when «= 0, i = le '■ . (c) i = ce~^' + -• If i = when t = 0, f = f (l - e~^' ). 218 ANSWERS. ((J) i = ce ^ -\ {B sin ut — Lw cos at). R (e) i = ce ^ -\ ^ — (B sin wt — La cos ut) + n, "^^,„ J -g sin (6o)J + e) - i6u cos (6(o« + »)}. t_ 35. (a) i = ^rj'e^/'(0c?« + cij1. (_ 1^ (6) i = ce "B. (c) i = ce "". t_ (d) i=ce ■*"+, — ^". ^ (cosat + BCasinut). 1 + Jc^ 07^(1)2 ~5o /• _L L 36. (a) (? =?— - I e^7(0* + ce *". _ { (6) q= Qe '"', wliere Q is the charge at time « = 0. t (c) g = C£ + ce~^. (d) g = ce ^' + -_^^__ (sin at-RCa cos «,«). 37. s^ '^^^^o'^+l-^ > CHAPTER VI. Art. SO. 3. a; = cie^' + cac-^. i. x= Ciel" + C2e~t'. Art. 51. 1. 3^ = e2»^(ci + cax) + Cse-"^. 2. ?/= e-'^Cd + C2a; + C8a;2)+ c^e^. Art. 52. 3. y = e''(ci + C2x)sin x + e'«(cs + C4a;)cos ». Art. 58. 2. 2/ = cie» + Cae-"^ - 2 - 5 a;. 3. ^ = e-'{cx + c^x) + - e^. 4. y = e^'Cci + e^) + cse-*" + e'* f f e-5» fe^'^'XCdx)'. ANSWERS. 219 Art. 60. 3. 3/ = e~2"'(cicos— a; + C2sin-— -a;j + e"^(cs + |a;)+ ^e^'- 1. 4. y = e'{ax+ 6)+|e?^ Art. 61. 3. j; = Cie-^ + e' (C2 cos V3 a; + Cs sin VSa;) + J(k* - » + 1). Art. 62. 3. ^ = C\e^ -\- CiB-'^ — r}^ sin J a;. 4. 1/ = ce~" + e^fcicos — ai + Casin — x\ sin 3 a; + 27 cos 3 a: 1 . sin x — cos x 730 2 4' r '^ Art. 63. 2. y = cxe-" + Cje-^ + -^^ (11 sin a; — 7 cos a;). 3. j/ = ccos(V2a; + o) + -^(lla;2-12a;+|5'\+|!(4sin2a;-cos2x) Art. 64. 2. j; = ciCos(2a; + o)+ ^aisina; — f cosa;. 3. ^ = CieC + 026-* + a; sin a; + I cosx(l — a;^). Page 80. \.. y = (fiiff + Cje"^) cos x + (cse"^ + 046""^) sin x. Z. y — cie-"' cos (a; + o) + c^e^" cos (2 a; + 0) + cse"^*. 3. 2/ = ci + e-«(c2 + C3a;) + |--5^ + 4a;+^. 4. 2/ = ciCOs2x+ C2sin2a; + ^(e=«-sin3a;)+i(2x2- 1). pfnx 5. 2/ = Cie2» + C2e3» + yV(6a; + 5) + 6. , = ci««« + c2e- + ^,-^+2„ 7. 2/ = Cie-2'^ + Cae^ + Cge" + ^x + i). »»2 — 5 TO + 6 a;e"' 220 AJSrSWERS. 8- V = ci + CiX + e ^fcs siu-^a; + d cos — a;^ + 3 fex^ - J ax' 9. y = cie' + c^e^ + cge-^"^ + I'l (a; + U). 10. y = c sin (na; + a) (aa: + 6) + /"^'"^^ ■ 11. 2/ = (a + 6x)sin x + (c + dx)cos x + ^1™^ + 9x^-xi ^^^ ^ 12 48 12. 2, = cicos(ax + a) + ^liM^ + ^25if^M^2^a?!. a a^ 13. y=(c + C2X)e»+^(2x2_4K + 3). _ 96(2a: + 3) __3i 15. 2/ = cie»« + 026-""^ + c^ sin (aa; + a) — «-*»« — 24 o-*. 16. J/ = ci + cax + e»^(c3 + c^x) + x^ + ^x^. 11. y = cxe" + Cid-" + Cs sin (x + a) — J e* cos x. ,V3_ , , „:„ V^ J- 18. y = e Vcicos— x + czsin— ia;]-,Jj(2cos2a; + 3sin2a;). 12 V 12 j p2i 19. 2/ = Cie-« + cie-^ + cae*" 20. 2/ = ce*sin(V3a; + a)+ ^e'^cosx. 21. y = e-''(ci + 62X + cgx^) + ^x'e-'. 22. y = e2='(ci + cax) + e-»(c3 + Ctx) + J e"^(x2 + 2 x + J). 23. 2/ = e'(ci + C2X + csx^ + J xs + jfij x4) . 24. !/ = cie" + cze-'^ — J(xsinx + oosx)+ -^xe''(2z^ — 8x'+ 9). 25. 2^ = Cie*" + Cze"^ — ^{2 sin 3 x + cos 3 x) — — (sin 2 x + cos 2 x). 8 26. y = e'(ci + C2 cos x + Cs sin x) + xe' + t^(cos x + 3 sin x) . 27. y = Cie5« + cje** + x + t^. ^8. 2/ = e2«(ci + cjx) + cae'''' + J e*». J X 29. 2/ = cie-"' + e [ ca cos — x + Ca sin x ). + .^ (3 sinx - 11 cos x) - I xeYsin^^ + VS cos^Y 130^ ^ ^ \ 2 2 I ANSWERS. 221 CHAPTER VII. Alt. 65. 'a/3, 2. y = cx^ cos [ — log a; + a ] + x''. Art. 66. 3. !/ = (c, + C2 log a;)sin log a; + (ca + c^ log a;)cos log x. Art. 69. 2. y = x-Hci + c^\ogx-) + ^- 3. !/=cia;-Hc2x*- g + | + iy Art. 71. 1. y = i5 + 2 x)2 [ci(5 + 2 a;)^2+ C2(5 + 2 x)-'^^]. 2. !/=(29;-l)[ci + C2(2a;-l)2 + C3(2a;-l) 2 ]. Page 91. 1. w = Cia;2 + a;2(c2a; >= + CsX 2 )- ^- 2- 2/ = V-i + CiX-" + x-H\ 5 %. y = X\Cx + C2 logx + C3(l0gx)2]. 4. 2/ = ci(x + a)2 + C2(x + a)3 + 3£j±l«. o i. y = x-\c\ + C2 logx) + C3X. 6. 2/ = x(ci cos log X + C2 sin log x + 5) + x-i(C8 + 2 log x) . r , 1 / . ,^-, /^-T . C3 + C 4log(x +1) , x" + 52x + 51 7. y = [ci + C2log(x + l)]Vx+l+-^5 * °^ ^ + ^jxs Vx + 1 '^■^'' x™ 8. y = CiX + C2X-i + ^^_^ 9. I/ = x2(Ci + C2l0gX) + ^^J^3^- 10. 2/ = C.F. of Ex. 3, Art. 66, + (log x)^ + 2 log x - 3. 11. y = x(ci + C2 log X) + csx-i + J x-i log X. 12. WzziflOg-^— +Cil0gX + C2). X\ X— 1 / 222 ANSWERS. 13. y = x'"{cism log k" + C2 cos log x" + log x) . 14. y = a;2(cia;*^ + Ci/c-^^) + „— + 4f- (5 sin log a; + 6 cos log x) X OJ. uC 2 + (27 sin logs; + 191 cos logx). CHAPTER VIII. Art. 75. 4. e^^'y = i(e^x«dx + CiCe'^dx + cj. 6. dy = V^y^T^dx. 7. aj/^ + CiK^ = cj. Art. 76. 2. 3/ = Ci + CaK + Csx^ H [- CbK"-' + 1=- — |m + m 3. y = ci + C2X + CaX^ + CiX^ + ^x^ log x. 4. y = Ci + C2X + (6 — a;2) sin a; — 4 k cos x. Art. 77. 2. 3 a; = 2 a*(!/* - 2 ci) (y* + ci)* + Cj. 3. Vci2/2 + y log( Vc^/ + Vl + 61^) = aciv^ a; + C2. 4. OK = log(^ + V^'^ + Ci) + C2, or y = ci'e""' + Cs'e-"*. Art. 78. 2. 2(2/-6) = e'-» + e-(»'-»). 3. j/ = cja; + (ci^ + 1) log (a; - ci) + Cj. 4. 16 cx^y = 4 (cia; + a^)i + CiX + cj. Art. 79. 1. e-"" = c-iX + C2. 3. log y = Cifi"' + c^e ". 2. y'' = x^ + CiX + Ci. 4. sin (ci — 2 V2 ^) = c^e'^. Art. 80. 1. y = ci sin ax + Cj cos ax + c^x + Cj. Z. y = Cie"" + Cae-"" + cs + 040; + cex^ + - a?(jofi - m") ANSWERS. 223 3. J^ = ci + C2S + a;^(c8a; ^ + C4a; ^ ) when a ^. Art. 81. 2. 2ci^ = ci2e°+e » + C2. 3. «/ = ciloga; + Ca. 4. 15 2/ = 8(a; + Cj)^ + C2X + cs. Art. 82. + ^-H^+A + 2tAt7 + 2.4.6'3.7.11 + -) 1.3 1.3.3.7 1.3. 6-3. 7 -11 r.2 6. 2/ = ^l[l-Ji.M + l.^+n-n-2.re + l.ji + 3.^ \ + B(x-n-l-n + 2^+n-l-n-S-n + 2.n + 4~ V V [3 ii / * \ 2.2n-l 2.4.2re-1.2re-3 ^ +^a;--i('l+"+^-^+^x-H''+'""+^-"+^-"+*a:-^+-V \, 2.2n+3 2.4.2»i + 3.2n+5 >/ Page 107. .'■ ^fx^^ay+^y-'^- ^- ^'S.+ ^fx+-'y =''+'=■ 2. 2ay + x^ = cW¥^^ + C2. S. 2/ = ci + Cja; + cae'" + Cie""". 3. (1 + a; + a;2)2/ = cia;2 + C2X + Ca. 6. Cit^ = C2e«=' - nVT+o^c?. 7. xyy/x'^ — 1 = sec-i a; + CiVa;^ - 1 + c^ log (a: + Vx^ - 1) + cj. 8. 2/ = C2 - sln-i Cie-». 9. y = ci sin-i a; + (sin-i xy + c^ 224 AN8WMBS. 10. y = ci sin2 a; + C2 cos x — C2 sin^x log tan -• 11. y = C2e"^(x — 1) + cix + ci fxe"' i x-^e-'^(^dx)^ + ca. 12. 2/ = axlogx + cix + c. ^4_ y=;xHc.x+cs+^+5Hl2x. 13. log,-l=— L_. 1^ 16 C1X+C2 15. 2/ = Cie-»' + C2 + ie»^- 16. j/ = e-="=^ re""»^(cix + C2)(?x + cae """^ smx- 1 2 17. !/ = ci(l -xcotx) + C2Cotx. 18. alog(j/ + 6) = x + c. 19. (CiX + 62)2 + O = Cl!/2. 21. y— 6 = — logsecaK(x — c). CHAPTER IX. Art. 87. 2. y=:Ax + Bx(x-^ei^dx + l. 3. 2/ = ^e» + JSe3»'(4a;8- 42x2+ 150 X- 183). 4. 2/ = - + cifx + -V X \ x/ Art. 88. 2 2 2 2 3. y = ae^ + Cze^'J e*« "'(j,. i. y = ae^ + cae^ fx-^e^'^'^dx. Art. 91. -— / /~ 2. 2/ = ce ^x^sin (— logx + o). 3. 2/ = (cisin -\/6x + C2COS V6x) secx. 4. y = e'(ciX^ + cix) Art. 92. 2. 2/ = csmY21ogtan|+ay 3. ?/ = cisin (x2 + 0) + - 4. 2/ = Ci cos-^+ Casin-^ H — ?—• " 2x2 2x2^ aV x'' AN S WEBS. 225 Page 120. 1. y = l(cie"'+C2e-«^). 4. »/ Vl + a;^ = Ci log (k + VI + x^) + Cj. X 2. xy = c sin (nx + a). 5. y = cje* + C2e'"^(4 x^ — 42x^ + 150 a; — 183) . S. !/ = ex sin ()ix + a). 6. y = ce^'"' sin {xVb + a). 1. y = e-'^iciS''^ + Cie-''^). 8. 2/ = e»(Ciloga; + C2). 9, 2/ = cix + C2(x sin-i x + Vl - x^) — i x(l - x^)^. 10. ^ = cix + C2 cos X. 11. 2/ = cix^ + C2X + Ca I x^ I x-'e-»^dx — x | x'h'^dxy 12. 2/ = Cie" ='"""'* + C2e-»=™~-'*. 13. 2/ = Cix + C2X ) x-^e ^dx + x | x-^e ' \ xe ^f{x) (dxy. 14. 2y = x(cxe2»= + C2 — x). 17. j/^ = cx^ + cix. 15. !/ = cism -vl ho. n < ^ J , 19. y = c cos — ■ . 16. y = ci sin (m Vx^ — 1 + o). \ x j CHAPTER X. Page 124. 1. The circle of radius — K 2. Acatenary, 2/ = -(e ' +e ' \. 3. y^ -\-{x — ay = cr', circles whose centres are on the x-axis. 4. (x — ay = 4 o{y — c), a system of parabolas whose axes are parallel to the axis of y. 5. X + Ci = c vers-i- — V2 c^ - y'^, the cycloid obtained by rolling any circle along the x-axis from any point. 6. The ellipses aV + t^(« - c)^ = «*> if the cube of the normal is — k2 times the radius of curvature. The hyperbolas a'^y^ - k^(x - cy = a*, if the cube of the normal is + K^ times the radius of curvature. A set of parabolas if no constant is introduced at the first integration. 226 ANSWERS. 7. The elastic curve represented by the equation {4 k2 - (a;2 - a2)2jj dy = ± (a;^ - a^)dx. 8. s = ciB"' + cae-K', when accel = k^ distance from the fixed point. s = ci sin (Kt + Ci), when accel =— k^ distance from the fixed point. 9. s = iat^ + »o< + So- lo. The relation between time of motion and the distance passed over is i = ci ± -^^^{-Vs^ — cs + c log (Vs + Vi^^)}, according as the ^2 acceleration is ±— . s2 11. s = — log cosh nt, if the resistance of the air is — times the square of n^ g the velocity. 12. s = — — " ° , if the acceleration is — k times the cube of the velocity. 13. T = 2Tr-\}-^- (Emtage, Mathematical Theory of Electricity and ^ MH Magnetism, p. 85.) 14. s=l + (so — V) cos Kt, V =— k(so — sin Kt, where k ■. Hint : Put s—l equal to a new variable. 2 16. -^, if acceleration is — -^ 17. i = - je ''ij"eV'(0*-e ^jfeV'CO*} t_ _t_ + Cie ^1 + Cae 'i, where Ti = ^^ ^ and T^ = - ^ ^^- 18. Same as in 17 with /(«) substituted tovf'(t). 19. i = e 2i(ci+C2«)- 20. i=:7sin — !: — Vie 21. e — o = Cie-'« sin (v'iu^-k^i+cj) for u > k ; 9-0. = cie-(«-^'<''-<"2)< + C2e-(''+^«^ "=)' for w< k. 2. 6 £/(/ = P(3 IH - a;3) . 23. 24 .B/j/ = «)(4 i^x - a;*). ANSWERS. 227 24. The general solution is . = ^coshV|.+ ^sinh^^. + || + ^^. On applying the conditions (Jf Ex. 22 to determine the constants, . _ WMI -, -B = 0; and therefore, y = ^^ ( 1 - cosh -y/^- x^ + ^ §2 V ^m / 2Q CHAPTER XI. Art. 98. 2. a; = e«'(^cos« + Bsint), y = e^[{A- B)cost + (A + B)smt'\. 3. a; = Cie-5' + C2e' + fe2'-|«-i|, y = -cie-^ -\- c^e' + ^e^ - it- ^l i. x = cie-'+C2e-«'+-V- «-¥—¥«'' 2/=-Cie-«+4c2e-6'— y-«+¥+¥«'- 5. x= (ci+C20e' + (C3 + C40e"'i 2j/=(C2-Ci-C20e'-(c8 + C4+C40e-'. Art. 99. 3. x^ = g^ + ci, x3 = 2,3 _|_ C2. 4. « = ;^^ log ^ + ci, y - X = cxy. y — X X 5. X^ — J/' = Ci', x' + - = C2. 6. ax2 + 6j/2 + c«2 = ci, a2x2 + h'^y'^ + c^z^ = cj. Art. 103. 3. (y + z)e' = e. i. x - cy -y log 2 = 0. 5, e»^(jf + z2 + x) = c. 6. y(x + z)=c(y + z). Page 143. 1. X = cie"?' + ^ e2' - A e', X + 1 = Cae"' + ie^- A e'. 2. X = (ci sin « + C2 cos «)« * + -^ — 7^- y=Ke2- ci) sin « - (C2 + ci) cos J]e-* - ^f- + Jy" 3. !/ = (ci + C2X) 6=^+3 Cse-^'' - J x, « = 2 (3 C2 - Ci - C2x) e»^ - Cse"!"' - f 4. X = cie-2' + cae-^' - ,^-5 + A« - i-e'. 228 ANSWERS. 9. log xyz + x + y + z = c. 6. x^+y^+e^=cu x^+y'—z^=Ci. 10. (M + z) (« + c) + z(k - m) = 0. '■ ~^'^~jr~ 11- x^ + a;2,2 - M + a;% = c. 12. z2 ^. (X _ a)2 + (!/ - 6)2 = ft2. X = ci cos V3 8 + C2 sin V3 « + cs cos V2 { + ci sin V2 « - tII^ e3' + ^V ies* - tV - i cos 2 f, 2/ = -3 ci cos V3 S - 3 C2 sin V3 { - 2 C3 cos V2 J - 2 C4 sin V2 1 13. 14. X = e^^fci cos M + c, sin MU e" ^2(03 cos ^+ a sin :^') . V V2 V2/ V V'2 v^/ 2/ = e^fci sin ^ - c. cos MU e" ^2f c, cos M _ C3 sin ^y 1-1/2 v'2/ V V2 v'2/ X = Oi sin kJ + 02 cos /ci+ 03, 15. y = 61 sin Kf + 62 cos Kt + 63, z = ci sin Ki + C2 cos xt + cs, where k^ = P + m' + n'' ; and the arbitrary constants are con- nected by the following relations : mci — n5i _ nai — Ici _ Ibi — mai _ j^ 02 t>2 C2 Zai + »»6i + «ci = 0,^ = ^ = ^. le. See Forsyth, Diff. Eq., Ex. 3, Art. 174 ; Johnson, Diff. Eq., Art. 242. 17. X + miy = cie(<'+'^''''*' + c2e-(''+'"i''''*', where mi and m^ are the roots of a'm^ +(a — b)m — 6 = 0. Ex. 16, p. 269, Johnson, Diff. Eq. ; Ex. 4, p. 270, Forsyth, Diff. Eq. 18. When the horizontal and vertical lines through the starting point in the plane of motion are taken for the x and y axes, the equation of the path is X = vot cos 4>, y = vdt sin — ^ gfi ; and the elimination of * gives the parabola y=xta-n(p — ig—- — — . va' cos-* 19. Axes being chosen as in Ex. 18, »; = Hocos 0(1 - e-"}, y = -^-t + ^"°""/ + ^ (1 - «"«). ANSWERS. 229 20. For upper sign : the hyperbola (ai!/ — 6ia;)(M— 02y) = (ai62—a2&i)''- For lower sign : the ellipse (aiy—bix)^ +(^a-iy—bixy=(aib2—aibi)\ CHAPTER XII. Art. 108. 2. z=px + qy + pq. 3. p = q. 4. q = 2yp^. 5. z = pq. ax» Vax^ dx dy^ \dyl dy Art. 109. 2. yp^xq = 0. 3. (? + »ip)^ + «(«g - »»P) = (w + nq)x. dx^^dy ^ dy' dx' Art. 116. 2. «=e»0(a;-J/). 3. ix + mt/ + rez = 0(a;'' + 2/'* + 2^)- 4. i_l=//'i_lV 5. /(a;2-22, a;S-!/')=0. X y \x z) Art. 117. 2. a:2,.-3„ = ^(|.^). Art. 119. 3 z=a{x + ^^zz^Vc. 4- « = ma; + 2/e"« + c V 2+VlO / 5. z = ax+^+h. a Art. 120. 2. z = aa; + by -iVab. Art. 121. 2. gVteJ+l-i _ j^2^a_ 3. (2 + a2)3 = (x + ay + c)". 4. 4c(3 - a) = (x + cy + B)2 + 4. 5. ^2 _[. [2Vz2-4o2 - 4 a^ log (» + Vi^^^Io^)] = 4(x + ay + 6). 230 ANSWERS. Art. 122. 2. at = (x + a)t + (!/ -I- a)t + 6. 4. s = ^(2 a; - a)^ + a^^ + 6. 3. z =ax + aPy^ + b. S. a = f(a;+ a)t + f (j( - a)t + 6. 6. g^ = gVa'' + g^ + y Vj/i' - a' + g^ log a: + Va?" + a'' ^ ^_ 2/ + V2/2 _ o2 Art. 123. 2. 2 = (a; + a) (^ + 6), S.I. is 3 = 0. Another form of the C.I. is 2Va = - + ay + 1). 3. C.I. is s2 = aV + (aa; + 6)*. Art. 125. 4. « = : i?(2/)a;» +f(y) Art. 126. 3. z = :^f(A. h^f^Y 3- a2 = ^ + a;0(2/)+f(?/). 2. X =/(»)+ 0(2/). Art. 128. 2. « = 0(3(-2x)+i/'(j/-x). 3. » = 0(2/ + 3x)+i^(^-2x). Art. 129. 1. S = X20i(2, + X) + X02(2/ + X) + 03(2/ + x). Art. 130. 12 ■■ 6 3. ^- 4. ^ + ji,x4. Art. 131. 4. s = 0(x + 2/)+e3i'i^(2/-x). 5. e = e''i>(y)+ e-'^ix + y) 6. z = er'^(y — x) + e-'^^{y + x). Art. 132. 3. -(j,e»+2y + ^ijx8 + |x2!/ + Jx2 + ^x2/ + ^\x), Jsin(x + 22/)-xc», - k&'-'i + ix22/ + ix2 + Jxy + f X + f 2/ + -V-. ANSWERS. 231 Art. 133. 2. (.x^y) + xi^(,x'h/) + ^- 3. z = 0(a;2 + 2,2) +i/(x^- y^). (Put x for i x^, y for ^ ^2.) Page 187. 2. a;2^.2,-2 4. 2-2 — 2^1 _ j. 6. alogg = aex +(1 - ac)y + b. 4 y-^ = ^l^^^Z^\. 8. z + Vx2 + 2/2 + ^2 = a;i-«0('^V ■ s-c Vs-cy • \x/ 9. J^^^::^, ?^l^\ = fi. 10. x + 2/ + « = 0(a;2/z). 11. {cos(a; + 2/) + sin(x + 2/)}e!'-» = .#. { z^tan (^ - £il) } • 12. VTT^ - logi+^^S^ = X + ay + 6. 13. a;2 = as^ + 2V^ + 6. ^g^ z = ax + ^ °' V + c. 14. (g-6-alogx)2 = 4a22/. n±Vn2_4 16. z = ax + 6j/ + cVl + a2 + 62. S.I. is x2 + 2/2 + z'^ = c2 17. 3 = ax + (1 - Va)'h/ + 6. 18. z = axes + 4 a^e^y + b. 19. az — 1 = 06"+"". 20. (1) 2z=(-+a2/y + &; (2) z = X2/ +2/ Vx2 - ai2 + 61 ; (3) z = xy + xVf+ a^ + 62. 21. 2 = I a log (x2 + 2/2) + Vn^ *™"'| + *• (C'hange to polar co-ordi- nates.) 22 AL/-l'=^!!iL+ riL. 2-i m + 1 « + l 23. z = av^Ti+ vT:^^-/^^ + &. (Put Vx + i/ = M, Vx-2/ = »■) 24. z = axy + a2(x + 2/) + 6- (P"t xy = v, x + ?/ = w.) 232 ANSWEBS. 25. Vl + as =2y/x + ay + b. 27. zy^ = xy + F(_x)+'p(y). 26. z = iifly^ + ^(y)losx + ^(y). 28. y = x{z)+^j/(z). 39. a; = 0Cz)+i/'(a; + y + z). 30. (n - l)s2/ + OK = e(''-i)*i|J'(j;) +f(y). 31. « = j'e-J>(='>*'reJ>W'^i;'(y)da; + 0(2;)l(ia; + ^(v). 32. z = iyx!^logx + x^(y-)+^(>(y). 33. « = I a^,, ^xy + (y)+ e-^(y). 34. iz = xV + Hy)+^{x). 35. s = - ^^** + ^ + -F(y - 6a;) + /(J' - ax)- 36. 2^ + x{ax + hy + cz) = iZ-Caa; +hy + cz). 37. 30=aa;(^2-l)(2/2 + 2)+0(a;)vT=r^^ (K2/). 38. ^z = (>xV+{i-'Y'Ji + yl'{y). INDEX OP NAMES. (The numbers refer to pages.) Airy, 205. Aldis, 150, 153. Bedell, 145. Bernoulli, 28. Bessell, 105, 106. Bocher, 207. Boole, 23, fi(i, 206. Boussinesq, 206. Briot and Bouquet, 193, 194, 203. Brooks, 204. Byerly, 105, 106, 169, 184, 205. Cajori, 202. Cauchy, 193, 194, 203. .Cayley, 40, 48. Charpit, 166. Chrystal, 49. Clairaut, 36, 40, 44. Craig, 202, 203, 204, 207. D'Alembert, 146, 173. De Martres, 200. De Morgan, 205. DuBois-Eeymond, 206. Duhamel, 206. Edwards, 4, 5, 48, 52, 54, 149, 183, 184, 205. Emtage, 126, 183, 185. Euler, 64, 107, 146. Fiske, 193. Forsyth, 48, 69, 80, 86, 94, 101, 106, 107, 111, 116, 138, 159, 168, 172, 202, 205, 206. Puchs, 203. Galois, 204. Gauss, 107, 202. Gilbert, 193. Glaisher, 48, 106. Goursat, 207. Gray, 106. Halphen, 204, 205. Heffter, 207. Hilhert, 190, 194. Hill, 49. Hoiiel, 206. Hymers, 206. Johnson, 25, 48, 80, 105, 106, 107, 111, 138, 149, 168, 176, 180, 205, 206. Jordan, 206. Klein, 207. Koenigsberger, 193, 206. Kowalevsky, 194. Lagrange, 40, 114, 151, 154, 155, 166. Lagnerre, 204. Lamb, 184. Laplace, 105, 182, 184. Laurent, 206. Legendre, 90, 105, 184. Leibniz, 27, 40. Lie, 202, 204, 207. Liouville, 200. Lipschitz, 193. Mansion, 193, 207. Mathews, 106, 203, 207. McMahon, 54, 65, 196, 197. Meray, 193. Merriman, 126, 127. Merriman and Woodward, 54, 127, 176, 202. 233 234 INDEX OF NAMES. Moigno, 25, 193, 206. Monge, 171. Newton, 27. Osborne, 205. Page, 204, 207. Painlevg, 206, 207. Peirce, 183, 186. Picard, 193, 19i, 206. Pockels, 207. Poinear^, 207. Poisson, 186. Price, 206. Riccati, 105, 106. Biemann, 203, 206. Schlesinger, 207. Serret, 206. Smith, C, 150, 153, 160. Smith, D. E., 202. Stegemann, 205. Taylor, 40. Thomson and Tait, 183, 185, 186. Todhunter, 106, 183. Weierstrass, 193, 203. Williamson, 4, 52, 149, 183, 184. INDEX OF SUBJECTS. (The numbers refer to pages.) Applications to geometry, 50-58, 121, 134,140,141,158. mechanics, 58, 122, 144. physics, 62, 122, 126, 144, 145. Auxiliary equation, 65-67, 175. Bessel's equation, 105, 106. Characteristic, 151. Charpit, method of, 166. Clairaut's equation, 36, 44. Complementary function, 63, 174, 179. Condition that equation he exact, 18, 92, 197. Constants of integration, 2, 149, 194, 195. Criterion for independence of, con- stants, 195. integrals, 197. of integrahility, 136, 138, 200. Cuspidal locus, 47. Derivation, of ordinary equations, 4. of partial equations, 146, 148. Discriminant, 40. Envelope, 41-44, 150. Equation, Equations. Auxiliary, 65-67, 175. Bessel's, 105, 106. Clairaut's, 36, 44. Decomposahle, 31. Definitions, 1, 17, 92, 146. Derivation of, 4, 146, 148. Exact, 17-19, 92-94, 197. Geometrical meaning, 8-10, 134, 140, 142, 158. Homogeneous, 15, 35, 82-84, 90, 138, 174. Equation, Equations. Invariants of, 204. Laplace's, 182. Legendre's, 90, 105. Linear, ordinary, 26, 28, 63, 64, 70, 82-84, 90, 101, 128. Linear, partial, 153-158, 173. Linear, simultaneous, 128-133. Monge's, 171. Non-homogeneous, 16, 179. not decomposahle, 32. of cuspidal locus, 47-49. of envelope, 42. of hypergeometric series, 105, 107. of nodal locus, 45, 48, 49. of second order, 109-119. of tac-locus, 45, 48, 49. Partial. Partial, definition, derivation, 2, 146, 148. Partial, linear, 153-158, 169-187. Partial, linear homogeneous, 174- 178. Partial, linear non-homogeneous, 179-181. Partial , non-linear, integrals of, 149- 153. Partial, of first order, 149-169. Partial, of second and higher orders, 169-187. Poisson's, 186. Reduction to equiyalentsystem,189. Eiccati's, 105, 106. Simultaneous, 128-134. Single integrable, 136, 138, 200. Single non-integrable, 142. Transformation of, 28, 90, 114, 115, 117, 182. Works on, 205-207. 235 236 INDEX OF SUBJECTS. Existence Theorem, 190. Factors, integrating, 21-26. Geometrical meaning, 8-10, 134, 140, 142, 158. Geometry. See applications. Homogeneous. See equation. Hypergeometric series, equation of, 105, 107. Integrability, criterioti of, 136, 138, 200. Integral, integrals ; also see solution. and coefficients, 199. complete, 64, 149. first, 94. general, 150. of linear partial equations, 153. of simultaneous equations, 133. particular, G, 64, 73-80, 87-90, 149, 176, 180. relation between. 111. singular, 150. Integration, constants of. See con- stants. Integrating factors, 21-26. Invariants, 204. Lagrange's solution, 154, 155. Lagrangean lines, 155. Laplace's equation, 182. Legendre's equation, 105. Locus, 8, 42, 45, 47-49, 141, 151. Mechanics, applications to. See appli- cations. Modem theories, 202. Monge's equations, method, 171. Nodal locus, 45, 48, 49. Particular integrals. See integral. Physics, applications to. See applica- tions. Poisson's equation, 186. Reduction of equations to equivalent system, 189. Relation between integrals. 111. Relation between integrals and coeffi- cients, 199. Removal of second term, 115. Riccati's equation, 105, 106. Series, equation of hypergeometric, 105, 107. integration in, 101. Solutions, 2, 6 ; also see integrals. Spherical harmonics, 183, 184. Standard forms, 159-165. Summary, 38, 48. Symbol J), 67, 68. Symbolic function — — , 70. Symbolic functions/(fl),-i^, 86. Symbolic functions /(D, D'), 174, 176. f{.D,m Tac-locus, 45, 48, 49. Theories, modern, 202. Trajectories, 55-57. Transformations. See equations. Works on differential equations, 205. 1^^